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Mirrors > Home > MPE Home > Th. List > ssconb | Structured version Visualization version Unicode version |
Description: Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.) |
Ref | Expression |
---|---|
ssconb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3426 |
. . . . . . 7
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2 | ssel 3426 |
. . . . . . 7
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3 | pm5.1 868 |
. . . . . . 7
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4 | 1, 2, 3 | syl2an 480 |
. . . . . 6
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5 | con2b 336 |
. . . . . . 7
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6 | 5 | a1i 11 |
. . . . . 6
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7 | 4, 6 | anbi12d 717 |
. . . . 5
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8 | jcab 874 |
. . . . 5
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9 | jcab 874 |
. . . . 5
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10 | 7, 8, 9 | 3bitr4g 292 |
. . . 4
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11 | eldif 3414 |
. . . . 5
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12 | 11 | imbi2i 314 |
. . . 4
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13 | eldif 3414 |
. . . . 5
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14 | 13 | imbi2i 314 |
. . . 4
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15 | 10, 12, 14 | 3bitr4g 292 |
. . 3
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16 | 15 | albidv 1767 |
. 2
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17 | dfss2 3421 |
. 2
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18 | dfss2 3421 |
. 2
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19 | 16, 17, 18 | 3bitr4g 292 |
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-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 |
This theorem depends on definitions: df-bi 189 df-an 373 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-v 3047 df-dif 3407 df-in 3411 df-ss 3418 |
This theorem is referenced by: pssdifcom1 3853 pssdifcom2 3854 sbthlem1 7682 sbthlem2 7683 rpnnen2lem11 14277 setscom 15153 dpjidcl 17691 clsval2 20065 regsep2 20392 ordtconlem1 28730 conss2 36796 |
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