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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3597 | . . . . . . 7 | |
2 | ssel 3597 | . . . . . . 7 | |
3 | pm5.1 902 | . . . . . . 7 | |
4 | 1, 2, 3 | syl2an 494 | . . . . . 6 |
5 | con2b 349 | . . . . . . 7 | |
6 | 5 | a1i 11 | . . . . . 6 |
7 | 4, 6 | anbi12d 747 | . . . . 5 |
8 | jcab 907 | . . . . 5 | |
9 | jcab 907 | . . . . 5 | |
10 | 7, 8, 9 | 3bitr4g 303 | . . . 4 |
11 | eldif 3584 | . . . . 5 | |
12 | 11 | imbi2i 326 | . . . 4 |
13 | eldif 3584 | . . . . 5 | |
14 | 13 | imbi2i 326 | . . . 4 |
15 | 10, 12, 14 | 3bitr4g 303 | . . 3 |
16 | 15 | albidv 1849 | . 2 |
17 | dfss2 3591 | . 2 | |
18 | dfss2 3591 | . 2 | |
19 | 16, 17, 18 | 3bitr4g 303 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wal 1481 wcel 1990 cdif 3571 wss 3574 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 |
This theorem is referenced by: pssdifcom1 4054 pssdifcom2 4055 sbthlem1 8070 sbthlem2 8071 rpnnen2lem11 14953 setscom 15903 dpjidcl 18457 clsval2 20854 regsep2 21180 ordtconnlem1 29970 conss2 38647 |
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