Theorem issetri 2772

Description: A way to say "𝐴 is a set" (inference form). (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
issetri.1	⊢ ∃𝑥 𝑥 = 𝐴

Assertion

Ref	Expression
issetri	⊢ 𝐴 ∈ V

Distinct variable group:   𝑥,𝐴

Proof of Theorem issetri

Step	Hyp	Ref	Expression
1		issetri.1	. 2 ⊢ ∃𝑥 𝑥 = 𝐴
2		isset 2769	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	mpbir 146	1 ⊢ 𝐴 ∈ V

Syntax hints:   = wceq 1364  ∃wex 1506   ∈ wcel 2167  Vcvv 2763

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765

This theorem is referenced by:  0ex  4160  inex1  4167  vpwex  4212  zfpair2  4243  uniex  4472  bdinex1  15545  bj-zfpair2  15556  bj-uniex  15563  bj-omex2  15623