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Theorem eubrv 46417
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 𝐴𝑅𝑏𝐴 ∈ V)
Distinct variable groups:   𝐴,𝑏   𝑅,𝑏

Proof of Theorem eubrv
StepHypRef Expression
1 brprcneu 6887 . 2 𝐴 ∈ V → ¬ ∃!𝑏 𝐴𝑅𝑏)
21con4i 114 1 (∃!𝑏 𝐴𝑅𝑏𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  ∃!weu 2558  Vcvv 3471   class class class wbr 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149
This theorem is referenced by:  eubrdm  46418  afv2eu  46618
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