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Just to show how dumb I am, I have to ask, does this implication follow from the ZFC axioms?
[tex]x=y\ \land\ y\in z\rightarrow x\in z[/tex]
It's obviously true if we use the meaning of the = symbol, i.e. that the formula "x=y" means that x and y are different symbols for the same set, but I thought proofs in ZFC (and other formal languages) shouldn't ever have to rely on the meaning of the symbols.
[tex]x=y\ \land\ y\in z\rightarrow x\in z[/tex]
It's obviously true if we use the meaning of the = symbol, i.e. that the formula "x=y" means that x and y are different symbols for the same set, but I thought proofs in ZFC (and other formal languages) shouldn't ever have to rely on the meaning of the symbols.