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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 6653 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | 1 | eqeq1i 2627 | . 2 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘〈𝐴, 𝐵〉) = ∅) |
3 | afvfv0bi 41232 | . 2 ⊢ ((𝐹‘〈𝐴, 𝐵〉) = ∅ ↔ ((𝐹'''〈𝐴, 𝐵〉) = ∅ ∨ (𝐹'''〈𝐴, 𝐵〉) = V)) | |
4 | df-aov 41198 | . . . . 5 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
5 | 4 | eqeq1i 2627 | . . . 4 ⊢ ( ((𝐴𝐹𝐵)) = ∅ ↔ (𝐹'''〈𝐴, 𝐵〉) = ∅) |
6 | 5 | bicomi 214 | . . 3 ⊢ ((𝐹'''〈𝐴, 𝐵〉) = ∅ ↔ ((𝐴𝐹𝐵)) = ∅) |
7 | 4 | eqeq1i 2627 | . . . 4 ⊢ ( ((𝐴𝐹𝐵)) = V ↔ (𝐹'''〈𝐴, 𝐵〉) = V) |
8 | 7 | bicomi 214 | . . 3 ⊢ ((𝐹'''〈𝐴, 𝐵〉) = V ↔ ((𝐴𝐹𝐵)) = V) |
9 | 6, 8 | orbi12i 543 | . 2 ⊢ (((𝐹'''〈𝐴, 𝐵〉) = ∅ ∨ (𝐹'''〈𝐴, 𝐵〉) = V) ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V)) |
10 | 2, 3, 9 | 3bitri 286 | 1 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∨ wo 383 = wceq 1483 Vcvv 3200 ∅c0 3915 〈cop 4183 ‘cfv 5888 (class class class)co 6650 '''cafv 41194 ((caov 41195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-dfat 41196 df-afv 41197 df-aov 41198 |
This theorem is referenced by: (None) |
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