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Theorem brrelex1 5582
 Description: If two classes are related by a binary relation, then the first class is a set. (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
brrelex1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)

Proof of Theorem brrelex1
StepHypRef Expression
1 brrelex12 5581 . 2 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21simpld 498 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2114  Vcvv 3469   class class class wbr 5042  Rel wrel 5537 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 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 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539 This theorem is referenced by:  brrelex1i  5585  posn  5614  frsn  5616  releldm  5791  relelrn  5792  relimasn  5930  funmo  6350  ertr  8291  dirtr  17837  eqvreltr  35960  frege129d  40394  nnfoctb  41615  clim2d  42254  climfv  42272  meadjiun  43044  caragenunicl  43102
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