Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  brrelex1 Structured version   Visualization version   GIF version

Theorem brrelex1 5394
 Description: A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (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 5393 . 2 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21simpld 490 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∈ wcel 2164  Vcvv 3414   class class class wbr 4875  Rel wrel 5351 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-rel 5353 This theorem is referenced by:  brrelex1i  5397  posn  5426  frsn  5428  releldm  5595  relelrn  5596  relimasn  5733  funmo  6143  ertr  8029  dirtr  17596  issetssr  34800  eqvreltr  34896  frege129d  38895  nnfoctb  40029  clim2d  40698  climfv  40716  meadjiun  41472  caragenunicl  41530
 Copyright terms: Public domain W3C validator