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Theorem brrelex2 5729
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 5727 . 2 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21simprd 494 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2104  Vcvv 3472   class class class wbr 5147  Rel wrel 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682
This theorem is referenced by:  brrelex2i  5732  releldm  5942  relelrn  5943  elrelimasn  6083  funbrfv  6941  relbrtpos  8224  ertr  8720  erth  8754  pslem  18529  opeldifid  32097  eqvreltr  37780  eqvrelth  37784  fsuppss  41371  frege124d  42814  frege133d  42818  climfv  44705  funbrafv2  46253
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