Theorem bj-isseti 34189

Description: Remove from isseti 3508 dependency on ax-ext 2793 (and on df-cleq 2814 and df-v 3496). This proof uses only df-clab 2800 and df-clel 2893 on top of first-order logic. It only uses ax-12 2173 among the auxiliary logical axioms. The hypothesis uses 𝑉 instead of V for extra generality. This is indeed more general as long as elex 3512 is not available. Use bj-issetiv 34188 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-isseti.1	⊢ 𝐴 ∈ 𝑉

Assertion

Ref	Expression
bj-isseti	⊢ ∃𝑥 𝑥 = 𝐴

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-isseti

Step	Hyp	Ref	Expression
1		bj-isseti.1	. 2 ⊢ 𝐴 ∈ 𝑉
2		bj-elisset 34187	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1533  ∃wex 1776   ∈ wcel 2110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-sb 2066  df-clab 2800  df-clel 2893

This theorem is referenced by:  bj-rexcom4b  34194