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Mirrors > Home > ILE Home > Th. List > disjpr2 | Unicode version |
Description: The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.) |
Ref | Expression |
---|---|
disjpr2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3567 | . . . 4 | |
2 | 1 | a1i 9 | . . 3 |
3 | 2 | ineq2d 3308 | . 2 |
4 | indi 3354 | . . 3 | |
5 | df-pr 3567 | . . . . . . . 8 | |
6 | 5 | ineq1i 3304 | . . . . . . 7 |
7 | indir 3356 | . . . . . . 7 | |
8 | 6, 7 | eqtri 2178 | . . . . . 6 |
9 | disjsn2 3622 | . . . . . . . . . 10 | |
10 | 9 | adantr 274 | . . . . . . . . 9 |
11 | 10 | adantr 274 | . . . . . . . 8 |
12 | disjsn2 3622 | . . . . . . . . . 10 | |
13 | 12 | adantl 275 | . . . . . . . . 9 |
14 | 13 | adantr 274 | . . . . . . . 8 |
15 | 11, 14 | jca 304 | . . . . . . 7 |
16 | un00 3440 | . . . . . . 7 | |
17 | 15, 16 | sylib 121 | . . . . . 6 |
18 | 8, 17 | syl5eq 2202 | . . . . 5 |
19 | 5 | ineq1i 3304 | . . . . . . 7 |
20 | indir 3356 | . . . . . . 7 | |
21 | 19, 20 | eqtri 2178 | . . . . . 6 |
22 | disjsn2 3622 | . . . . . . . . . 10 | |
23 | 22 | adantr 274 | . . . . . . . . 9 |
24 | 23 | adantl 275 | . . . . . . . 8 |
25 | disjsn2 3622 | . . . . . . . . . 10 | |
26 | 25 | adantl 275 | . . . . . . . . 9 |
27 | 26 | adantl 275 | . . . . . . . 8 |
28 | 24, 27 | jca 304 | . . . . . . 7 |
29 | un00 3440 | . . . . . . 7 | |
30 | 28, 29 | sylib 121 | . . . . . 6 |
31 | 21, 30 | syl5eq 2202 | . . . . 5 |
32 | 18, 31 | uneq12d 3262 | . . . 4 |
33 | un0 3427 | . . . 4 | |
34 | 32, 33 | eqtrdi 2206 | . . 3 |
35 | 4, 34 | syl5eq 2202 | . 2 |
36 | 3, 35 | eqtrd 2190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wne 2327 cun 3100 cin 3101 c0 3394 csn 3560 cpr 3561 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-sn 3566 df-pr 3567 |
This theorem is referenced by: (None) |
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