Theorem aovov0bi 48207

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 7415	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2	1	eqeq1i 2766	. 2 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
3		afvfv0bi 48163	. 2 ⊢ ((𝐹‘⟨𝐴, 𝐵⟩) = ∅ ↔ ((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ∨ (𝐹'''⟨𝐴, 𝐵⟩) = V))
4		df-aov 48132	. . . . 5 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
5	4	eqeq1i 2766	. . . 4 ⊢ ( ((𝐴𝐹𝐵)) = ∅ ↔ (𝐹'''⟨𝐴, 𝐵⟩) = ∅)
6	5	bicomi 227	. . 3 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ↔ ((𝐴𝐹𝐵)) = ∅)
7	4	eqeq1i 2766	. . . 4 ⊢ ( ((𝐴𝐹𝐵)) = V ↔ (𝐹'''⟨𝐴, 𝐵⟩) = V)
8	7	bicomi 227	. . 3 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) = V ↔ ((𝐴𝐹𝐵)) = V)
9	6, 8	orbi12i 928	. 2 ⊢ (((𝐹'''⟨𝐴, 𝐵⟩) = ∅ ∨ (𝐹'''⟨𝐴, 𝐵⟩) = V) ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V))
10	2, 3, 9	3bitri 300	1 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V))

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861   = wceq 1570  Vcvv 3451  ∅c0 4279  ⟨cop 4590  ‘cfv 6531  (class class class)co 7412  '''cafv 48128   ((caov 48129

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415  df-aiota 48096  df-dfat 48130  df-afv 48131  df-aov 48132

This theorem is used by: (None)