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Metamath Proof Explorer

Theorem ax6dgen

Description: Tarski's system uses the weaker ax6v instead of the bundled ax-6 , so here we show that the degenerate case of ax-6 can be derived. Even though ax-6 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 is ax6v . We later rederive from ax6v the bundled form as ax6 with the help of the auxiliary axiom schemes. (Contributed by NM, 23-Apr-2017)

		Ref	Expression
	Assertion	ax6dgen	⊢ ¬ ∀ 𝑥 ¬ 𝑥 = 𝑥

Proof

Step	Hyp	Ref	Expression
1		equid	⊢ 𝑥 = 𝑥
2	1	notnoti	⊢ ¬ ¬ 𝑥 = 𝑥
3	2	spfalw	⊢ ( ∀ 𝑥 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥 )
4	1 3	mt2	⊢ ¬ ∀ 𝑥 ¬ 𝑥 = 𝑥