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Theorem aovov0bi 42050
Description: The operation's value on an ordered pair is the empty set if and only if the alternative value of the operation on this ordered pair is either the empty set or the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
aovov0bi ((𝐴𝐹𝐵) = ∅ ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V))

Proof of Theorem aovov0bi
StepHypRef Expression
1 df-ov 6881 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
21eqeq1i 2804 . 2 ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
3 afvfv0bi 42006 . 2 ((𝐹‘⟨𝐴, 𝐵⟩) = ∅ ↔ ((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ∨ (𝐹'''⟨𝐴, 𝐵⟩) = V))
4 df-aov 41975 . . . . 5 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
54eqeq1i 2804 . . . 4 ( ((𝐴𝐹𝐵)) = ∅ ↔ (𝐹'''⟨𝐴, 𝐵⟩) = ∅)
65bicomi 216 . . 3 ((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ↔ ((𝐴𝐹𝐵)) = ∅)
74eqeq1i 2804 . . . 4 ( ((𝐴𝐹𝐵)) = V ↔ (𝐹'''⟨𝐴, 𝐵⟩) = V)
87bicomi 216 . . 3 ((𝐹'''⟨𝐴, 𝐵⟩) = V ↔ ((𝐴𝐹𝐵)) = V)
96, 8orbi12i 939 . 2 (((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ∨ (𝐹'''⟨𝐴, 𝐵⟩) = V) ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V))
102, 3, 93bitri 289 1 ((𝐴𝐹𝐵) = ∅ ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wo 874   = wceq 1653  Vcvv 3385  c0 4115  cop 4374  cfv 6101  (class class class)co 6878  '''cafv 41971   ((caov 41972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-int 4668  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-res 5324  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-aiota 41934  df-dfat 41973  df-afv 41974  df-aov 41975
This theorem is referenced by: (None)
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