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

Theorem brrelex12i 5457
Description: Two classes that are related by a binary relation are sets. (An artifact of our ordered pair definition.) (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 5454 . 2 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
31, 2mpan 677 1 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wcel 2050  Vcvv 3416   class class class wbr 4929  Rel wrel 5412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-xp 5413  df-rel 5414
This theorem is referenced by:  nprrel12  5460  vtoclr  5465  relbrcnvg  5808  ovprc  7013  oprabv  7033  encv  8314  fsuppimp  8634  fsuppunbi  8649  brfi1uzind  13667  isstruct2  16349  brssc  16942  isfull  17038  isfth  17042  dvdsr  19119  ulmval  24671  subgrv  26755  vcex  28132  opelco3  32535  bj-elep  33866  rngoablo2  34626  aovprc  42791  aovrcl  42792  nelbrim  42878  isisomgr  43355  linindsv  43865
  Copyright terms: Public domain W3C validator