Theorem eubrv 46996

Description: If there is a unique set which is related to a class, then the class must be a set. (Contributed by AV, 25-Aug-2022.)

Assertion

Ref	Expression
eubrv	⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V)

Distinct variable groups:   𝐴,𝑏   𝑅,𝑏

Proof of Theorem eubrv

Step	Hyp	Ref	Expression
1		brprcneu 6901	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑏 𝐴𝑅𝑏)
2	1	con4i 114	1 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2107  ∃!weu 2567  Vcvv 3479   class class class wbr 5149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5313  ax-pr 5439

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-ss 3981  df-nul 4341  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5150

This theorem is referenced by:  eubrdm  46997  afv2eu  47199