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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 3261 | . . . 4 | |
2 | eqtr 2182 | . . . . . 6 | |
3 | 2 | eqcomd 2170 | . . . . 5 |
4 | difeq1 3228 | . . . . . 6 | |
5 | difun2 3483 | . . . . . 6 | |
6 | eqtr 2182 | . . . . . . 7 | |
7 | incom 3309 | . . . . . . . . . 10 | |
8 | 7 | eqeq1i 2172 | . . . . . . . . 9 |
9 | disj3 3456 | . . . . . . . . 9 | |
10 | 8, 9 | bitri 183 | . . . . . . . 8 |
11 | eqtr 2182 | . . . . . . . . . 10 | |
12 | 11 | expcom 115 | . . . . . . . . 9 |
13 | 12 | eqcoms 2167 | . . . . . . . 8 |
14 | 10, 13 | sylbi 120 | . . . . . . 7 |
15 | 6, 14 | syl5com 29 | . . . . . 6 |
16 | 4, 5, 15 | sylancl 410 | . . . . 5 |
17 | 3, 16 | syl 14 | . . . 4 |
18 | 1, 17 | mpan 421 | . . 3 |
19 | 18 | com12 30 | . 2 |
20 | 19 | adantl 275 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 cdif 3108 cun 3109 cin 3110 wss 3111 c0 3404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 |
This theorem is referenced by: (None) |
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