Theorem eubrv 44529

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 6764	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑏 𝐴𝑅𝑏)
2	1	con4i 114	1 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2106  ∃!weu 2568  Vcvv 3432   class class class wbr 5074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075

This theorem is referenced by:  eubrdm  44530  afv2eu  44730