Theorem isseti 2782

Description: A way to say "𝐴 is a set" (inference form). (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
isseti.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
isseti	⊢ ∃𝑥 𝑥 = 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem isseti

Step	Hyp	Ref	Expression
1		isseti.1	. 2 ⊢ 𝐴 ∈ V
2		isset 2780	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	mpbi 145	1 ⊢ ∃𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1373  ∃wex 1516   ∈ wcel 2177  Vcvv 2773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-v 2775

This theorem is referenced by:  rexcom4b  2799  ceqsex  2812  ceqsexv2d  2814  vtoclf  2828  vtocl2  2830  vtocl3  2831  vtoclef  2850  eqvinc  2900  euind  2964  opabm  4335  eusv2nf  4511  dtruex  4615  limom  4670  isarep2  5370  dfoprab2  6005  rnoprab  6041  dmaddpq  7512  dmmulpq  7513  bj-inf2vnlem1  16044