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Mathbox for Alexander van der Vekens |
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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 7415 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | 1 | eqeq1i 2736 | . 2 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘〈𝐴, 𝐵〉) = ∅) |
3 | afvfv0bi 46319 | . 2 ⊢ ((𝐹‘〈𝐴, 𝐵〉) = ∅ ↔ ((𝐹'''〈𝐴, 𝐵〉) = ∅ ∨ (𝐹'''〈𝐴, 𝐵〉) = V)) | |
4 | df-aov 46288 | . . . . 5 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
5 | 4 | eqeq1i 2736 | . . . 4 ⊢ ( ((𝐴𝐹𝐵)) = ∅ ↔ (𝐹'''〈𝐴, 𝐵〉) = ∅) |
6 | 5 | bicomi 223 | . . 3 ⊢ ((𝐹'''〈𝐴, 𝐵〉) = ∅ ↔ ((𝐴𝐹𝐵)) = ∅) |
7 | 4 | eqeq1i 2736 | . . . 4 ⊢ ( ((𝐴𝐹𝐵)) = V ↔ (𝐹'''〈𝐴, 𝐵〉) = V) |
8 | 7 | bicomi 223 | . . 3 ⊢ ((𝐹'''〈𝐴, 𝐵〉) = V ↔ ((𝐴𝐹𝐵)) = V) |
9 | 6, 8 | orbi12i 912 | . 2 ⊢ (((𝐹'''〈𝐴, 𝐵〉) = ∅ ∨ (𝐹'''〈𝐴, 𝐵〉) = V) ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V)) |
10 | 2, 3, 9 | 3bitri 297 | 1 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 844 = wceq 1540 Vcvv 3473 ∅c0 4322 〈cop 4634 ‘cfv 6543 (class class class)co 7412 '''cafv 46284 ((caov 46285 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-res 5688 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-aiota 46252 df-dfat 46286 df-afv 46287 df-aov 46288 |
This theorem is referenced by: (None) |
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