Theorem aovov0bi 47205

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

Step	Hyp	Ref	Expression
1		df-ov 7413	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2	1	eqeq1i 2741	. 2 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
3		afvfv0bi 47161	. 2 ⊢ ((𝐹‘⟨𝐴, 𝐵⟩) = ∅ ↔ ((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ∨ (𝐹'''⟨𝐴, 𝐵⟩) = V))
4		df-aov 47130	. . . . 5 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
5	4	eqeq1i 2741	. . . 4 ⊢ ( ((𝐴𝐹𝐵)) = ∅ ↔ (𝐹'''⟨𝐴, 𝐵⟩) = ∅)
6	5	bicomi 224	. . 3 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ↔ ((𝐴𝐹𝐵)) = ∅)
7	4	eqeq1i 2741	. . . 4 ⊢ ( ((𝐴𝐹𝐵)) = V ↔ (𝐹'''⟨𝐴, 𝐵⟩) = V)
8	7	bicomi 224	. . 3 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) = V ↔ ((𝐴𝐹𝐵)) = V)
9	6, 8	orbi12i 914	. 2 ⊢ (((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ∨ (𝐹'''⟨𝐴, 𝐵⟩) = V) ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V))
10	2, 3, 9	3bitri 297	1 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V))

Syntax hints:   ↔ wb 206   ∨ wo 847   = wceq 1540  Vcvv 3464  ∅c0 4313  ⟨cop 4612  ‘cfv 6536  (class class class)co 7410  '''cafv 47126   ((caov 47127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413  df-aiota 47094  df-dfat 47128  df-afv 47129  df-aov 47130

This theorem is referenced by: (None)