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Mirrors > Home > NFE Home > Th. List > uneqdifeq | Unicode version |
Description: Two ways to say that and partition (when and don't overlap and is a part of ). (Contributed by FL, 17-Nov-2008.) |
Ref | Expression |
---|---|
uneqdifeq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3408 | . . . . 5 | |
2 | eqtr 2370 | . . . . . . 7 | |
3 | 2 | eqcomd 2358 | . . . . . 6 |
4 | difeq1 3246 | . . . . . . 7 | |
5 | difun2 3629 | . . . . . . 7 | |
6 | eqtr 2370 | . . . . . . . 8 | |
7 | incom 3448 | . . . . . . . . . . 11 | |
8 | 7 | eqeq1i 2360 | . . . . . . . . . 10 |
9 | disj3 3595 | . . . . . . . . . 10 | |
10 | 8, 9 | bitri 240 | . . . . . . . . 9 |
11 | eqtr 2370 | . . . . . . . . . . 11 | |
12 | 11 | expcom 424 | . . . . . . . . . 10 |
13 | 12 | eqcoms 2356 | . . . . . . . . 9 |
14 | 10, 13 | sylbi 187 | . . . . . . . 8 |
15 | 6, 14 | syl5com 26 | . . . . . . 7 |
16 | 4, 5, 15 | sylancl 643 | . . . . . 6 |
17 | 3, 16 | syl 15 | . . . . 5 |
18 | 1, 17 | mpan 651 | . . . 4 |
19 | 18 | com12 27 | . . 3 |
20 | 19 | adantl 452 | . 2 |
21 | difss 3393 | . . . . . . . 8 | |
22 | sseq1 3292 | . . . . . . . . 9 | |
23 | unss 3437 | . . . . . . . . . . 11 | |
24 | 23 | biimpi 186 | . . . . . . . . . 10 |
25 | 24 | expcom 424 | . . . . . . . . 9 |
26 | 22, 25 | syl6bi 219 | . . . . . . . 8 |
27 | 21, 26 | mpi 16 | . . . . . . 7 |
28 | 27 | com12 27 | . . . . . 6 |
29 | 28 | adantr 451 | . . . . 5 |
30 | 29 | imp 418 | . . . 4 |
31 | eqimss 3323 | . . . . . . 7 | |
32 | 31 | adantl 452 | . . . . . 6 |
33 | ssundif 3633 | . . . . . 6 | |
34 | 32, 33 | sylibr 203 | . . . . 5 |
35 | 34 | adantlr 695 | . . . 4 |
36 | 30, 35 | eqssd 3289 | . . 3 |
37 | 36 | ex 423 | . 2 |
38 | 20, 37 | impbid 183 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wceq 1642 cdif 3206 cun 3207 cin 3208 wss 3257 c0 3550 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 |
This theorem is referenced by: (None) |
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