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Nov 27, 2023 at 22:39 comment added D.S. Lipham @AndyPutman The one in the comment above is probably the simplest and I would accept it if written as an answer here. But I suspect there are many others that are less elementary but use established (well-known) results. That is the nature of my proof.
Nov 27, 2023 at 22:29 comment added Andy Putman You were already given an answer by The Amplitwist in the comments. Also, this is basically circular. The proof that $K_{3,3}$ is not planar uses the same kinds of things that go into proving the Jordan curve theorem type results that are needed to prove what you're trying to prove.
Nov 27, 2023 at 22:09 history edited D.S. Lipham CC BY-SA 4.0
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Nov 27, 2023 at 22:03 history answered D.S. Lipham CC BY-SA 4.0