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Mirrors > Home > ILE Home > Th. List > uneqdifeqim | Unicode version |
Description: Two ways that and can "partition" (when and don't overlap and is a part of ). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.) |
Ref | Expression |
---|---|
uneqdifeqim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3087 | . . . 4 | |
2 | eqtr 2057 | . . . . . 6 | |
3 | 2 | eqcomd 2045 | . . . . 5 |
4 | difeq1 3055 | . . . . . 6 | |
5 | difun2 3302 | . . . . . 6 | |
6 | eqtr 2057 | . . . . . . 7 | |
7 | incom 3129 | . . . . . . . . . 10 | |
8 | 7 | eqeq1i 2047 | . . . . . . . . 9 |
9 | disj3 3272 | . . . . . . . . 9 | |
10 | 8, 9 | bitri 173 | . . . . . . . 8 |
11 | eqtr 2057 | . . . . . . . . . 10 | |
12 | 11 | expcom 109 | . . . . . . . . 9 |
13 | 12 | eqcoms 2043 | . . . . . . . 8 |
14 | 10, 13 | sylbi 114 | . . . . . . 7 |
15 | 6, 14 | syl5com 26 | . . . . . 6 |
16 | 4, 5, 15 | sylancl 392 | . . . . 5 |
17 | 3, 16 | syl 14 | . . . 4 |
18 | 1, 17 | mpan 400 | . . 3 |
19 | 18 | com12 27 | . 2 |
20 | 19 | adantl 262 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 cdif 2914 cun 2915 cin 2916 wss 2917 c0 3224 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 |
This theorem is referenced by: (None) |
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