Theorem isseti 2824

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 2822	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	mpbi 145	1 ⊢ ∃𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1398  ∃wex 1541   ∈ wcel 2205  Vcvv 2815

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 2216

This theorem depends on definitions:  df-bi 117  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-v 2817

This theorem is referenced by:  rexcom4b  2841  ceqsex  2854  ceqsexv2d  2856  vtoclf  2870  vtocl2  2872  vtocl3  2873  vtoclef  2892  eqvinc  2943  euind  3007  opabm  4404  eusv2nf  4582  dtruex  4686  limom  4741  isarep2  5448  dfoprab2  6108  rnoprab  6144  dmaddpq  7710  dmmulpq  7711  bj-inf2vnlem1  16866