Theorem issetri 3314

Description: A way to say "𝐴 is a set" (inference rule). (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 3311	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	mpbir 221	1 ⊢ 𝐴 ∈ V

Syntax hints:   = wceq 1596  ∃wex 1817   ∈ wcel 2103  Vcvv 3304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-12 2160  ax-ext 2704

This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1599  df-ex 1818  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-v 3306

This theorem is referenced by:  zfrep4  4887  0ex  4898  inex1  4907  pwex  4953  zfpair2  5012  uniex  7070  bj-snsetex  33178