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Theorem aov0ov0 43742
Description: If the alternative value of the operation on an ordered pair is the empty set, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
aov0ov0 ( ((𝐴𝐹𝐵)) = ∅ → (𝐴𝐹𝐵) = ∅)

Proof of Theorem aov0ov0
StepHypRef Expression
1 afv0fv0 43698 . 2 ((𝐹'''⟨𝐴, 𝐵⟩) = ∅ → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
2 df-aov 43670 . . 3 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
32eqeq1i 2806 . 2 ( ((𝐴𝐹𝐵)) = ∅ ↔ (𝐹'''⟨𝐴, 𝐵⟩) = ∅)
4 df-ov 7142 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
54eqeq1i 2806 . 2 ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
61, 3, 53imtr4i 295 1 ( ((𝐴𝐹𝐵)) = ∅ → (𝐴𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  c0 4246  cop 4534  cfv 6328  (class class class)co 7139  '''cafv 43666   ((caov 43667
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-res 5535  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-aiota 43635  df-dfat 43668  df-afv 43669  df-aov 43670
This theorem is referenced by: (None)
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