ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  brrelex GIF version

Theorem brrelex 4678
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
brrelex ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)

Proof of Theorem brrelex
StepHypRef Expression
1 brrelex12 4676 . 2 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21simpld 112 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2158  Vcvv 2749   class class class wbr 4015  Rel wrel 4643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645
This theorem is referenced by:  releldm  4874  relelrn  4875  funmo  5243  ertr  6563
  Copyright terms: Public domain W3C validator