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Theorem brrelex12i 5571
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 5568 . 2 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
31, 2mpan 689 1 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2111  Vcvv 3441   class class class wbr 5030  Rel wrel 5524
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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
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 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526
This theorem is referenced by:  nprrel12  5574  vtoclr  5579  relbrcnvg  5935  ovprc  7173  oprabv  7193  encv  8500  fsuppimp  8823  fsuppunbi  8838  brfi1uzind  13852  brfi1indALT  13854  isstruct2  16485  brssc  17076  isfull  17172  isfth  17176  dvdsr  19392  ulmval  24975  subgrv  27060  vcex  28361  opelco3  33131  bj-ideqgALT  34573  bj-idreseqb  34578  bj-ideqg1ALT  34580  rngoablo2  35347  aovprc  43742  aovrcl  43743  nelbrim  43829  isisomgr  44340  linindsv  44852
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