Theorem isseti 2812

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 2810	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	mpbi 145	1 ⊢ ∃𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803

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 2213

This theorem depends on definitions:  df-bi 117  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2805

This theorem is referenced by:  rexcom4b  2829  ceqsex  2842  ceqsexv2d  2844  vtoclf  2858  vtocl2  2860  vtocl3  2861  vtoclef  2880  eqvinc  2930  euind  2994  opabm  4381  eusv2nf  4559  dtruex  4663  limom  4718  isarep2  5424  dfoprab2  6078  rnoprab  6114  dmaddpq  7642  dmmulpq  7643  bj-inf2vnlem1  16669