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Theorem brrelex2 5191
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 5189 . 2 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21simprd 478 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  Vcvv 3231   class class class wbr 4685  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150
This theorem is referenced by:  brrelex2i  5193  releldm  5390  relelrn  5391  elrelimasn  5524  funbrfv  6272  relbrtpos  7408  ertr  7802  erth  7834  pslem  17253  opeldifid  29538  frege124d  38370  frege133d  38374  climfv  40241
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