Theorem eubrv 47800

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 6871	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑏 𝐴𝑅𝑏)
2	1	con4i 115	1 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  ∃!weu 2595  Vcvv 3454   class class class wbr 5108

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109

This theorem is used by:  eubrdm  47801  afv2eu  48003