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Mirrors > Home > ILE Home > Th. List > diftpsn3 | Unicode version |
Description: Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.) |
Ref | Expression |
---|---|
diftpsn3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 3578 | . . . 4 | |
2 | 1 | a1i 9 | . . 3 |
3 | 2 | difeq1d 3234 | . 2 |
4 | difundir 3370 | . . 3 | |
5 | 4 | a1i 9 | . 2 |
6 | df-pr 3577 | . . . . . . . . 9 | |
7 | 6 | a1i 9 | . . . . . . . 8 |
8 | 7 | ineq1d 3317 | . . . . . . 7 |
9 | incom 3309 | . . . . . . . . 9 | |
10 | indi 3364 | . . . . . . . . 9 | |
11 | 9, 10 | eqtri 2185 | . . . . . . . 8 |
12 | 11 | a1i 9 | . . . . . . 7 |
13 | necom 2418 | . . . . . . . . . . 11 | |
14 | disjsn2 3633 | . . . . . . . . . . 11 | |
15 | 13, 14 | sylbi 120 | . . . . . . . . . 10 |
16 | 15 | adantr 274 | . . . . . . . . 9 |
17 | necom 2418 | . . . . . . . . . . 11 | |
18 | disjsn2 3633 | . . . . . . . . . . 11 | |
19 | 17, 18 | sylbi 120 | . . . . . . . . . 10 |
20 | 19 | adantl 275 | . . . . . . . . 9 |
21 | 16, 20 | uneq12d 3272 | . . . . . . . 8 |
22 | unidm 3260 | . . . . . . . 8 | |
23 | 21, 22 | eqtrdi 2213 | . . . . . . 7 |
24 | 8, 12, 23 | 3eqtrd 2201 | . . . . . 6 |
25 | disj3 3456 | . . . . . 6 | |
26 | 24, 25 | sylib 121 | . . . . 5 |
27 | 26 | eqcomd 2170 | . . . 4 |
28 | difid 3472 | . . . . 5 | |
29 | 28 | a1i 9 | . . . 4 |
30 | 27, 29 | uneq12d 3272 | . . 3 |
31 | un0 3437 | . . 3 | |
32 | 30, 31 | eqtrdi 2213 | . 2 |
33 | 3, 5, 32 | 3eqtrd 2201 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wne 2334 cdif 3108 cun 3109 cin 3110 c0 3404 csn 3570 cpr 3571 ctp 3572 |
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-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-sn 3576 df-pr 3577 df-tp 3578 |
This theorem is referenced by: (None) |
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