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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afvfv0bi | Structured version Visualization version Unicode version |
Description: The function's value at an argument is the empty set if and only if the value of the alternative function at this argument is either the empty set or the universe. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
afvfv0bi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioran 493 |
. . . 4
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2 | df-ne 2624 |
. . . . . . 7
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3 | afvnufveq 38649 |
. . . . . . 7
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4 | 2, 3 | sylbir 217 |
. . . . . 6
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5 | eqeq1 2455 |
. . . . . . . 8
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6 | 5 | notbid 296 |
. . . . . . 7
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7 | 6 | biimpd 211 |
. . . . . 6
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8 | 4, 7 | syl 17 |
. . . . 5
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9 | 8 | impcom 432 |
. . . 4
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10 | 1, 9 | sylbi 199 |
. . 3
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11 | 10 | con4i 134 |
. 2
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12 | afv0fv0 38651 |
. . 3
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13 | afvpcfv0 38648 |
. . 3
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14 | 12, 13 | jaoi 381 |
. 2
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15 | 11, 14 | impbii 191 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3047 df-sbc 3268 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-br 4403 df-opab 4462 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-res 4846 df-iota 5546 df-fun 5584 df-fv 5590 df-dfat 38617 df-afv 38618 |
This theorem is referenced by: aovov0bi 38698 |
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