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Theorem brcodir 5420
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 5197 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧𝑧𝑅𝐵)))
2 vex 3175 . . . . . 6 𝑧 ∈ V
3 brcnvg 5212 . . . . . 6 ((𝑧 ∈ V ∧ 𝐵𝑊) → (𝑧𝑅𝐵𝐵𝑅𝑧))
42, 3mpan 701 . . . . 5 (𝐵𝑊 → (𝑧𝑅𝐵𝐵𝑅𝑧))
54anbi2d 735 . . . 4 (𝐵𝑊 → ((𝐴𝑅𝑧𝑧𝑅𝐵) ↔ (𝐴𝑅𝑧𝐵𝑅𝑧)))
65adantl 480 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑅𝑧𝑧𝑅𝐵) ↔ (𝐴𝑅𝑧𝐵𝑅𝑧)))
76exbidv 1836 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑧(𝐴𝑅𝑧𝑧𝑅𝐵) ↔ ∃𝑧(𝐴𝑅𝑧𝐵𝑅𝑧)))
81, 7bitrd 266 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧𝐵𝑅𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wex 1694  wcel 1976  Vcvv 3172   class class class wbr 4577  ccnv 5026  ccom 5031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-cnv 5035  df-co 5036
This theorem is referenced by:  codir  5421
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