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Theorem brrelex2 5404
Description: A true binary relation on a relation implies the second argument is a set. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
brrelex2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)

Proof of Theorem brrelex2
StepHypRef Expression
1 brrelex12 5402 . 2 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21simprd 491 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2107  Vcvv 3398   class class class wbr 4886  Rel wrel 5360
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 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
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-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362
This theorem is referenced by:  brrelex2i  5407  releldm  5604  relelrn  5605  elrelimasn  5743  funbrfv  6493  relbrtpos  7645  ertr  8041  erth  8073  pslem  17592  opeldifid  29975  eqvreltr  34977  eqvrelth  34981  frege124d  39010  frege133d  39014  climfv  40831  funbrafv2  42288
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