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Mirrors > Home > MPE Home > Th. List > Mathboxes > aovov0bi | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
aovov0bi | ⊢ ((𝐴𝐹𝐵) = ∅ ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7159 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | 1 | eqeq1i 2826 | . 2 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘〈𝐴, 𝐵〉) = ∅) |
3 | afvfv0bi 43371 | . 2 ⊢ ((𝐹‘〈𝐴, 𝐵〉) = ∅ ↔ ((𝐹'''〈𝐴, 𝐵〉) = ∅ ∨ (𝐹'''〈𝐴, 𝐵〉) = V)) | |
4 | df-aov 43340 | . . . . 5 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
5 | 4 | eqeq1i 2826 | . . . 4 ⊢ ( ((𝐴𝐹𝐵)) = ∅ ↔ (𝐹'''〈𝐴, 𝐵〉) = ∅) |
6 | 5 | bicomi 226 | . . 3 ⊢ ((𝐹'''〈𝐴, 𝐵〉) = ∅ ↔ ((𝐴𝐹𝐵)) = ∅) |
7 | 4 | eqeq1i 2826 | . . . 4 ⊢ ( ((𝐴𝐹𝐵)) = V ↔ (𝐹'''〈𝐴, 𝐵〉) = V) |
8 | 7 | bicomi 226 | . . 3 ⊢ ((𝐹'''〈𝐴, 𝐵〉) = V ↔ ((𝐴𝐹𝐵)) = V) |
9 | 6, 8 | orbi12i 911 | . 2 ⊢ (((𝐹'''〈𝐴, 𝐵〉) = ∅ ∨ (𝐹'''〈𝐴, 𝐵〉) = V) ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V)) |
10 | 2, 3, 9 | 3bitri 299 | 1 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∨ wo 843 = wceq 1537 Vcvv 3494 ∅c0 4291 〈cop 4573 ‘cfv 6355 (class class class)co 7156 '''cafv 43336 ((caov 43337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-res 5567 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-aiota 43305 df-dfat 43338 df-afv 43339 df-aov 43340 |
This theorem is referenced by: (None) |
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