Theorem eubrv 46291

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 6872	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑏 𝐴𝑅𝑏)
2	1	con4i 114	1 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2098  ∃!weu 2554  Vcvv 3466   class class class wbr 5139

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 2695  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140

This theorem is referenced by:  eubrdm  46292  afv2eu  46492