This is an instance of a much more general result. (See Visser for an overview of various related principles.) A theory is called sequential if it supports encoding of sequences of its objects with some basic properties. As a part of the definition (which I omit here as it is technical and not particularly relevant, it can be found in Pudlák, see Visser for more discussion), a sequential theory has some designated natural numbers (which serve as lengths of sequences) defined by a predicate $N(x)$. Usual theories of sets or classes are sequential, with $N(x)$ being $x\in\omega$.
Theorem: For any sequential theory $T$, the following are equivalent:
1. $T$ proves full induction: the schema $$\forall\bar y\,[\varphi(0,\bar y)\land\forall x\,(N(x)\land\varphi(x,\bar y)\to\varphi(x+1,\bar y))\to\forall x\,(N(x)\to\varphi(x,\bar y))]$$ for all formulas $\varphi$.
2. $T$ is uniformly essentially reflexive: for every formula $\varphi(x)$ and a finite subtheory $S\subseteq T$, $T$ proves $N(x)\land\Pr_S(\ulcorner\varphi(\dot x)\urcorner)\to\varphi(x)$N(x)\land\Pr_S(\left\ulcorner\varphi(\dot x)\right\urcorner)\to\varphi(x)$, where$\Pr_S$denotes the provability predicate for$S$, and$\dot x$the numeral for$x$. MK proves full induction, since it has induction for subsets of$\omega$, and the full comprehension schema implies guarantees that any property of natural numbers defined by a formula actually defines a subset of$\omega$. (Notice that this fails for NBG: due to the restrictions on its comprehension schema, NBG in general cannot prove induction for formulas with class quantifiers.) Thus, MK is uniformly essentially reflexive. In particular, if we take$0\ne0$(with no occurrence of$x$) for$\varphi$, we see that MK proves$\neg\Pr_S(\left\ulcorner0\ne0\right\urcorner)$, i.e.,$\mathrm{Con}_S$, for every its finite subtheory$S$, such as$S=\mathrm{NBG}$. The main idea of the proof of$1\to2$(which goes back to Montague) is that using sequence encoding, one can give partial truth definitions (i.e., truth definitions for any finite set of formulas including their substitution instances). Reasoning within the theory, if$S$proves$\varphi$, then by the cut-elimination theorem, it has a sequent proof where each formula is a subformula of something in$S$or$\varphi$. Using a partial truth definition for this finite set of formulas, one proves by induction on the length of proof that all sequents in the proof are true, hence also$\varphi$holds. 2 added 140 characters in body This is an instance of a much more general result. (See Visser for an overview of various related principles.) A theory is called sequential if it supports encoding of sequences of its objects with some basic properties. As a part of the definition (which I omit here as it is technical and not particularly relevant, it can be found in Pudlák), see Visser for more discussion), a sequential theory has some designated natural numbers (which serve as lengths of sequences) defined by a predicate$N(x)$. Usual theories of sets or classes are sequential, with$N(x)$being$x\in\omega$. Theorem: For any sequential theory$T$, the following are equivalent: 1.$T$proves full induction: the schema $$\forall\bar y\,(\varphi(0,\bar y\,[\varphi(0,\bar y)\land\forall x\,(N(x)\land\varphi(x,\bar y)\to\varphi(x+1,\bar y))\to\forall x\,(N(x)\to\varphi(x,\bar y))$$ y))]$$for all formulas \varphi. 2. T is uniformly essentially reflexive: for every formula \varphi(x) and a finite subtheory S\subseteq T, T proves N(x)\land\Pr_S(\ulcorner\varphi(\dot x)\urcorner)\to\varphi(x), where \Pr_S denotes the provability predicate for S, and \dot x the numeral for x. MK proves full induction, since it has induction for subsets of \omega, and the full comprehension schema implies that any property of natural numbers defined by a formula actually defines a subset of \omega. The main idea of the proof of 1\to2 (which goes back to Montague) is that using sequence encoding, one can give partial truth definitions (i.e., truth definitions for any finite set of formulas including their substitution instances). Reasoning within the theory, if S proves \varphi, then by the cut-elimination theorem, it has a sequent proof where each formula is a subformula of something in S or \varphi. Using a partial truth definition for this finite set of formulas, one proves by induction on the length of proof that all sequents in the proof are true, hence also \varphi holds. 1 This is an instance of a much more general result. (See Visser for an overview of various related principles.) A theory is called sequential if it supports encoding of sequences of its objects with some basic properties. As a part of the definition (which I omit here as it is technical and not particularly relevant, it can be found in Pudlák), a sequential theory has some designated natural numbers (which serve as lengths of sequences) defined by a predicate N(x). Usual theories of sets or classes are sequential, with N(x) being x\in\omega. Theorem: For any sequential theory T, the following are equivalent: 1. T proves full induction: the schema$$\forall\bar y\,(\varphi(0,\bar y)\land\forall x\,(N(x)\land\varphi(x,\bar y)\to\varphi(x+1,\bar y))\to\forall x\,(N(x)\to\varphi(x,\bar y))$$for all formulas$\varphi$. 2.$T$is uniformly essentially reflexive: for every formula$\varphi(x)$and a finite subtheory$S\subseteq T$,$T$proves$N(x)\land\Pr_S(\ulcorner\varphi(\dot x)\urcorner)\to\varphi(x)$, where$\Pr_S$denotes the provability predicate for$S$, and$\dot x$the numeral for$x$. MK proves full induction, since it has induction for subsets of$\omega$, and the full comprehension schema implies that any property of natural numbers defined by a formula actually defines a subset of$\omega$. The main idea of the proof of$1\to2$is that using sequence encoding, one can give partial truth definitions (i.e., truth definitions for any finite set of formulas including their substitution instances). Reasoning within the theory, if$S$proves$\varphi$, then by the cut-elimination theorem, it has a sequent proof where each formula is a subformula of something in$S$or$\varphi$. Using a partial truth definition for this finite set of formulas, one proves by induction on the length of proof that all sequents in the proof are true, hence also$\varphi\$ holds.