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Theorem aovov0bi 41796
 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 6796 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
21eqeq1i 2776 . 2 ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
3 afvfv0bi 41752 . 2 ((𝐹‘⟨𝐴, 𝐵⟩) = ∅ ↔ ((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ∨ (𝐹'''⟨𝐴, 𝐵⟩) = V))
4 df-aov 41718 . . . . 5 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
54eqeq1i 2776 . . . 4 ( ((𝐴𝐹𝐵)) = ∅ ↔ (𝐹'''⟨𝐴, 𝐵⟩) = ∅)
65bicomi 214 . . 3 ((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ↔ ((𝐴𝐹𝐵)) = ∅)
74eqeq1i 2776 . . . 4 ( ((𝐴𝐹𝐵)) = V ↔ (𝐹'''⟨𝐴, 𝐵⟩) = V)
87bicomi 214 . . 3 ((𝐹'''⟨𝐴, 𝐵⟩) = V ↔ ((𝐴𝐹𝐵)) = V)
96, 8orbi12i 900 . 2 (((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ∨ (𝐹'''⟨𝐴, 𝐵⟩) = V) ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V))
102, 3, 93bitri 286 1 ((𝐴𝐹𝐵) = ∅ ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∨ wo 836   = wceq 1631  Vcvv 3351  ∅c0 4063  ⟨cop 4322  ‘cfv 6031  (class class class)co 6793  '''cafv 41714   ((caov 41715 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-dfat 41716  df-afv 41717  df-aov 41718 This theorem is referenced by: (None)
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