Theorem issetri 3490

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 3486	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	mpbir 230	1 ⊢ 𝐴 ∈ V

Syntax hints:   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475

This theorem is referenced by:  zfrep4  5300  0ex  5311  inex1  5321  vpwex  5381  zfpair2  5434  vuniex  7750  bj-snsetex  36475