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Theorem aovov0bi 43272
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 7148 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
21eqeq1i 2823 . 2 ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
3 afvfv0bi 43228 . 2 ((𝐹‘⟨𝐴, 𝐵⟩) = ∅ ↔ ((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ∨ (𝐹'''⟨𝐴, 𝐵⟩) = V))
4 df-aov 43197 . . . . 5 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
54eqeq1i 2823 . . . 4 ( ((𝐴𝐹𝐵)) = ∅ ↔ (𝐹'''⟨𝐴, 𝐵⟩) = ∅)
65bicomi 225 . . 3 ((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ↔ ((𝐴𝐹𝐵)) = ∅)
74eqeq1i 2823 . . . 4 ( ((𝐴𝐹𝐵)) = V ↔ (𝐹'''⟨𝐴, 𝐵⟩) = V)
87bicomi 225 . . 3 ((𝐹'''⟨𝐴, 𝐵⟩) = V ↔ ((𝐴𝐹𝐵)) = V)
96, 8orbi12i 908 . 2 (((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ∨ (𝐹'''⟨𝐴, 𝐵⟩) = V) ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V))
102, 3, 93bitri 298 1 ((𝐴𝐹𝐵) = ∅ ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wo 841   = wceq 1528  Vcvv 3492  c0 4288  cop 4563  cfv 6348  (class class class)co 7145  '''cafv 43193   ((caov 43194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-aiota 43162  df-dfat 43195  df-afv 43196  df-aov 43197
This theorem is referenced by: (None)
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