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Theorem eubrv 42092
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 6438 . 2 𝐴 ∈ V → ¬ ∃!𝑏 𝐴𝑅𝑏)
21con4i 114 1 (∃!𝑏 𝐴𝑅𝑏𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  ∃!weu 2585  Vcvv 3397   class class class wbr 4886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-nul 5025  ax-pow 5077
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887
This theorem is referenced by:  eubrdm  42093  afv2eu  42272
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