Theorem issetri 3455

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 3450	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	mpbir 231	1 ⊢ 𝐴 ∈ V

Syntax hints:   = wceq 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438

This theorem is referenced by:  zfrep4  5229  0ex  5243  inex1  5253  vpwex  5313  zfpair2  5369  vuniex  7672  bj-snsetex  37005