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Theorem brcodir 5974
Description: Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)
Assertion
Ref Expression
brcodir ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧𝐵𝑅𝑧)))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝑅   𝑧,𝑉   𝑧,𝑊

Proof of Theorem brcodir
StepHypRef Expression
1 brcog 5732 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧𝑧𝑅𝐵)))
2 vex 3498 . . . . . 6 𝑧 ∈ V
3 brcnvg 5745 . . . . . 6 ((𝑧 ∈ V ∧ 𝐵𝑊) → (𝑧𝑅𝐵𝐵𝑅𝑧))
42, 3mpan 688 . . . . 5 (𝐵𝑊 → (𝑧𝑅𝐵𝐵𝑅𝑧))
54anbi2d 630 . . . 4 (𝐵𝑊 → ((𝐴𝑅𝑧𝑧𝑅𝐵) ↔ (𝐴𝑅𝑧𝐵𝑅𝑧)))
65adantl 484 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑅𝑧𝑧𝑅𝐵) ↔ (𝐴𝑅𝑧𝐵𝑅𝑧)))
76exbidv 1918 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑧(𝐴𝑅𝑧𝑧𝑅𝐵) ↔ ∃𝑧(𝐴𝑅𝑧𝐵𝑅𝑧)))
81, 7bitrd 281 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧𝐵𝑅𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wex 1776  wcel 2110  Vcvv 3495   class class class wbr 5059  ccnv 5549  ccom 5554
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-cnv 5558  df-co 5559
This theorem is referenced by:  codir  5975
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