Theorem bj-isseti

Description: Remove from isseti dependency on ax-ext (and on df-cleq and df-v ). This proof uses only df-clab and df-clel on top of first-order logic. It only uses ax-12 among the auxiliary logical axioms. The hypothesis uses V instead of _V for extra generality. This is indeed more general as long as elex is not available. Use bj-issetiv instead when sufficient (in particular when V is substituted for _V ). (Contributed by BJ, 13-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	bj-isseti.1	⊢ 𝐴 ∈ 𝑉
Assertion	bj-isseti	⊢ ∃ 𝑥 𝑥 = 𝐴

Proof

Step	Hyp	Ref	Expression
1		bj-isseti.1	⊢ 𝐴 ∈ 𝑉
2		bj-elisset	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
3	1 2	ax-mp	⊢ ∃ 𝑥 𝑥 = 𝐴