Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aovpcov0 | Structured version Visualization version GIF version |
Description: If the alternative value of the operation on an ordered pair is the universal class, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aovpcov0 | ⊢ ( ((𝐴𝐹𝐵)) = V → (𝐴𝐹𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | afvpcfv0 43344 | . 2 ⊢ ((𝐹'''〈𝐴, 𝐵〉) = V → (𝐹‘〈𝐴, 𝐵〉) = ∅) | |
2 | df-aov 43319 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
3 | 2 | eqeq1i 2826 | . 2 ⊢ ( ((𝐴𝐹𝐵)) = V ↔ (𝐹'''〈𝐴, 𝐵〉) = V) |
4 | df-ov 7158 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
5 | 4 | eqeq1i 2826 | . 2 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘〈𝐴, 𝐵〉) = ∅) |
6 | 1, 3, 5 | 3imtr4i 294 | 1 ⊢ ( ((𝐴𝐹𝐵)) = V → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Vcvv 3494 ∅c0 4290 〈cop 4572 ‘cfv 6354 (class class class)co 7155 '''cafv 43315 ((caov 43316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-int 4876 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-res 5566 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-aiota 43284 df-dfat 43317 df-afv 43318 df-aov 43319 |
This theorem is referenced by: (None) |
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