Theorem isseti 2779

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 2777	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	mpbi 145	1 ⊢ ∃𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1372  ∃wex 1514   ∈ wcel 2175  Vcvv 2771

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-v 2773

This theorem is referenced by:  rexcom4b  2796  ceqsex  2809  ceqsexv2d  2811  vtoclf  2825  vtocl2  2827  vtocl3  2828  vtoclef  2845  eqvinc  2895  euind  2959  opabm  4326  eusv2nf  4502  dtruex  4606  limom  4661  isarep2  5360  dfoprab2  5991  rnoprab  6027  dmaddpq  7491  dmmulpq  7492  bj-inf2vnlem1  15839