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Theorem brrelex2 5127
 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 5125 . 2 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21simprd 479 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1987  Vcvv 3190   class class class wbr 4623  Rel wrel 5089 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091 This theorem is referenced by:  brrelex2i  5129  releldm  5328  relelrn  5329  elrelimasn  5458  funbrfv  6201  relbrtpos  7323  ertr  7717  erth  7751  pslem  17146  opeldifid  29298  frege124d  37573  frege133d  37577  climfv  39359
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