Theorem eubrv 48027

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 6863	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑏 𝐴𝑅𝑏)
2	1	con4i 115	1 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ∃!weu 2593  Vcvv 3450   class class class wbr 5102

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103

This theorem is used by:  eubrdm  48028  afv2eu  48230