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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 Unicode 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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 6323 |
. . 3
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2 | 1 | eqeq1i 2467 |
. 2
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3 | afvfv0bi 38789 |
. 2
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4 | df-aov 38754 |
. . . . 5
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5 | 4 | eqeq1i 2467 |
. . . 4
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6 | 5 | bicomi 207 |
. . 3
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7 | 4 | eqeq1i 2467 |
. . . 4
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8 | 7 | bicomi 207 |
. . 3
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9 | 6, 8 | orbi12i 528 |
. 2
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10 | 2, 3, 9 | 3bitri 279 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4541 ax-nul 4550 ax-pow 4598 ax-pr 4656 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-rab 2758 df-v 3059 df-sbc 3280 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-br 4419 df-opab 4478 df-id 4771 df-xp 4862 df-rel 4863 df-cnv 4864 df-co 4865 df-dm 4866 df-res 4868 df-iota 5569 df-fun 5607 df-fv 5613 df-ov 6323 df-dfat 38752 df-afv 38753 df-aov 38754 |
This theorem is referenced by: (None) |
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