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Theorem brcodir 6139
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 5877 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧𝑧𝑅𝐵)))
2 vex 3484 . . . . . 6 𝑧 ∈ V
3 brcnvg 5890 . . . . . 6 ((𝑧 ∈ V ∧ 𝐵𝑊) → (𝑧𝑅𝐵𝐵𝑅𝑧))
42, 3mpan 690 . . . . 5 (𝐵𝑊 → (𝑧𝑅𝐵𝐵𝑅𝑧))
54anbi2d 630 . . . 4 (𝐵𝑊 → ((𝐴𝑅𝑧𝑧𝑅𝐵) ↔ (𝐴𝑅𝑧𝐵𝑅𝑧)))
65adantl 481 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑅𝑧𝑧𝑅𝐵) ↔ (𝐴𝑅𝑧𝐵𝑅𝑧)))
76exbidv 1921 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑧(𝐴𝑅𝑧𝑧𝑅𝐵) ↔ ∃𝑧(𝐴𝑅𝑧𝐵𝑅𝑧)))
81, 7bitrd 279 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧𝐵𝑅𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wex 1779  wcel 2108  Vcvv 3480   class class class wbr 5143  ccnv 5684  ccom 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-co 5694
This theorem is referenced by:  codir  6140
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