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Theorem brrelex2 5593
Description: If two classes are related by a binary relation, then the second class is a set. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
brrelex2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)

Proof of Theorem brrelex2
StepHypRef Expression
1 brrelex12 5591 . 2 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21simprd 499 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  Vcvv 3480   class class class wbr 5052  Rel wrel 5547
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 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
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-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549
This theorem is referenced by:  brrelex2i  5596  releldm  5801  relelrn  5802  elrelimasn  5940  funbrfv  6704  relbrtpos  7893  ertr  8294  erth  8328  pslem  17812  opeldifid  30353  eqvreltr  35912  eqvrelth  35916  frege124d  40315  frege133d  40319  climfv  42196  funbrafv2  43666
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