Theorem isseti 2822

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 2820	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	mpbi 145	1 ⊢ ∃𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1398  ∃wex 1541   ∈ wcel 2203  Vcvv 2813

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-v 2815

This theorem is referenced by:  rexcom4b  2839  ceqsex  2852  ceqsexv2d  2854  vtoclf  2868  vtocl2  2870  vtocl3  2871  vtoclef  2890  eqvinc  2940  euind  3004  opabm  4399  eusv2nf  4577  dtruex  4681  limom  4736  isarep2  5443  dfoprab2  6100  rnoprab  6136  dmaddpq  7694  dmmulpq  7695  bj-inf2vnlem1  16740