This is a follow-up question to http://mathoverflow.net/questions/230260/does-x-times-0-cup-x-times-1-leq-x-for-x-infinite-imply-s

Consider the statements

>> $(\text{S}1)$ For any infinite set $X$ there is an injection $\varphi$ from $(X\cup\{X\})$ into $X$.

and

>> $(\text{S}2)$ For any infinite set $X$ there is an injection $\varphi$ from $(X\times\{0\}) \cup (X\times\{1\})$ into $X$.

Is it true that ${\sf ZF} < {\sf ZF}+(\text{S1}) < {\sf ZF}+(\text{S}2)$? (Here $<$ means "strictly weaker".)