Theorem issetri 3450

Description: A way to say "𝐴 is a set" (inference form). (Contributed by NM, 21-Jun-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 3445	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	mpbir 232	1 ⊢ 𝐴 ∈ V

Syntax hints:   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433

This theorem is referenced by:  zfrep4  5215  0ex  5229  inex1  5245  vpwex  5306  zfpair2  5363  prex  5367  vuniex  7682  bj-snsetex  37316