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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 3535 | . . . 4 | |
2 | 1 | a1i 9 | . . 3 |
3 | 2 | difeq1d 3193 | . 2 |
4 | difundir 3329 | . . 3 | |
5 | 4 | a1i 9 | . 2 |
6 | df-pr 3534 | . . . . . . . . 9 | |
7 | 6 | a1i 9 | . . . . . . . 8 |
8 | 7 | ineq1d 3276 | . . . . . . 7 |
9 | incom 3268 | . . . . . . . . 9 | |
10 | indi 3323 | . . . . . . . . 9 | |
11 | 9, 10 | eqtri 2160 | . . . . . . . 8 |
12 | 11 | a1i 9 | . . . . . . 7 |
13 | necom 2392 | . . . . . . . . . . 11 | |
14 | disjsn2 3586 | . . . . . . . . . . 11 | |
15 | 13, 14 | sylbi 120 | . . . . . . . . . 10 |
16 | 15 | adantr 274 | . . . . . . . . 9 |
17 | necom 2392 | . . . . . . . . . . 11 | |
18 | disjsn2 3586 | . . . . . . . . . . 11 | |
19 | 17, 18 | sylbi 120 | . . . . . . . . . 10 |
20 | 19 | adantl 275 | . . . . . . . . 9 |
21 | 16, 20 | uneq12d 3231 | . . . . . . . 8 |
22 | unidm 3219 | . . . . . . . 8 | |
23 | 21, 22 | syl6eq 2188 | . . . . . . 7 |
24 | 8, 12, 23 | 3eqtrd 2176 | . . . . . 6 |
25 | disj3 3415 | . . . . . 6 | |
26 | 24, 25 | sylib 121 | . . . . 5 |
27 | 26 | eqcomd 2145 | . . . 4 |
28 | difid 3431 | . . . . 5 | |
29 | 28 | a1i 9 | . . . 4 |
30 | 27, 29 | uneq12d 3231 | . . 3 |
31 | un0 3396 | . . 3 | |
32 | 30, 31 | syl6eq 2188 | . 2 |
33 | 3, 5, 32 | 3eqtrd 2176 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wne 2308 cdif 3068 cun 3069 cin 3070 c0 3363 csn 3527 cpr 3528 ctp 3529 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-sn 3533 df-pr 3534 df-tp 3535 |
This theorem is referenced by: (None) |
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