Theorem isseti 2694

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 2692	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	mpbi 144	1 ⊢ ∃𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688

This theorem is referenced by:  rexcom4b  2711  ceqsex  2724  vtoclf  2739  vtocl2  2741  vtocl3  2742  vtoclef  2759  eqvinc  2808  euind  2871  opabm  4202  eusv2nf  4377  dtruex  4474  limom  4527  isarep2  5210  dfoprab2  5818  rnoprab  5854  dmaddpq  7187  dmmulpq  7188  bj-inf2vnlem1  13168