MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brrelex12i Structured version   Visualization version   GIF version

Theorem brrelex12i 5744
Description: Two classes that are related by a binary relation are sets. Inference form. (Contributed by BJ, 3-Oct-2022.)
Hypothesis
Ref Expression
brrelexi.1 Rel 𝑅
Assertion
Ref Expression
brrelex12i (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem brrelex12i
StepHypRef Expression
1 brrelexi.1 . 2 Rel 𝑅
2 brrelex12 5741 . 2 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
31, 2mpan 690 1 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  Vcvv 3478   class class class wbr 5148  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696
This theorem is referenced by:  nprrel12  5747  vtoclr  5752  relbrcnvg  6126  ovprc  7469  oprabv  7493  encv  8992  brdomi  8998  domssl  9037  fsuppimp  9406  fsuppunbi  9427  brttrcl  9751  brfi1uzind  14544  brfi1indALT  14546  isstruct2  17183  brssc  17862  isfull  17964  isfth  17968  dvdsr  20379  ulmval  26438  subgrv  29302  vcex  30607  opelco3  35756  bj-ideqgALT  37141  bj-idreseqb  37146  bj-ideqg1ALT  37148  rngoablo2  37896  aovprc  47138  aovrcl  47139  nelbrim  47225  linindsv  48291
  Copyright terms: Public domain W3C validator