Theorem bj-issetiv 35757

Description: Version of bj-isseti 35758 with a disjoint variable condition on 𝑥, 𝑉. The hypothesis uses 𝑉 instead of V for extra generality. This is indeed more general than isseti 3490 as long as elex 3493 is not available (and the non-dependence of bj-issetiv 35757 on special properties of the universal class V is obvious). Prefer its use over bj-isseti 35758 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-issetiv.1	⊢ 𝐴 ∈ 𝑉

Assertion

Ref	Expression
bj-issetiv	⊢ ∃𝑥 𝑥 = 𝐴

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Proof of Theorem bj-issetiv

Step	Hyp	Ref	Expression
1		bj-issetiv.1	. 2 ⊢ 𝐴 ∈ 𝑉
2		elissetv 2815	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1542  ∃wex 1782   ∈ wcel 2107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-clel 2811

This theorem is referenced by:  bj-rexcom4bv  35762  bj-vtoclf  35795