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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aov0ov0 | Structured version Visualization version GIF version |
Description: If the alternative value of the operation on an ordered pair is the empty set, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aov0ov0 | ⊢ ( ((𝐴𝐹𝐵)) = ∅ → (𝐴𝐹𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | afv0fv0 46411 | . 2 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) = ∅ → (𝐹‘⟨𝐴, 𝐵⟩) = ∅) | |
2 | df-aov 46383 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩) | |
3 | 2 | eqeq1i 2731 | . 2 ⊢ ( ((𝐴𝐹𝐵)) = ∅ ↔ (𝐹'''⟨𝐴, 𝐵⟩) = ∅) |
4 | df-ov 7407 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩) | |
5 | 4 | eqeq1i 2731 | . 2 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) = ∅) |
6 | 1, 3, 5 | 3imtr4i 292 | 1 ⊢ ( ((𝐴𝐹𝐵)) = ∅ → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∅c0 4317 ⟨cop 4629 ‘cfv 6536 (class class class)co 7404 '''cafv 46379 ((caov 46380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-res 5681 df-iota 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-aiota 46347 df-dfat 46381 df-afv 46382 df-aov 46383 |
This theorem is referenced by: (None) |
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