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Theorem eubrv 43420
 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 6635 . 2 𝐴 ∈ V → ¬ ∃!𝑏 𝐴𝑅𝑏)
21con4i 114 1 (∃!𝑏 𝐴𝑅𝑏𝐴 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2115  ∃!weu 2653  Vcvv 3471   class class class wbr 5039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-nul 5183  ax-pow 5239 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040 This theorem is referenced by:  eubrdm  43421  afv2eu  43587
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