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Mirrors > Home > ILE Home > Th. List > ssrelrel | Unicode version |
Description: A subclass relationship determined by ordered triples. Use relrelss 5035 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
ssrelrel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3061 | . . . 4 | |
2 | 1 | alrimiv 1830 | . . 3 |
3 | 2 | alrimivv 1831 | . 2 |
4 | elvvv 4572 | . . . . . . . 8 | |
5 | eleq1 2180 | . . . . . . . . . . . . . 14 | |
6 | eleq1 2180 | . . . . . . . . . . . . . 14 | |
7 | 5, 6 | imbi12d 233 | . . . . . . . . . . . . 13 |
8 | 7 | biimprcd 159 | . . . . . . . . . . . 12 |
9 | 8 | alimi 1416 | . . . . . . . . . . 11 |
10 | 19.23v 1839 | . . . . . . . . . . 11 | |
11 | 9, 10 | sylib 121 | . . . . . . . . . 10 |
12 | 11 | 2alimi 1417 | . . . . . . . . 9 |
13 | 19.23vv 1840 | . . . . . . . . 9 | |
14 | 12, 13 | sylib 121 | . . . . . . . 8 |
15 | 4, 14 | syl5bi 151 | . . . . . . 7 |
16 | 15 | com23 78 | . . . . . 6 |
17 | 16 | a2d 26 | . . . . 5 |
18 | 17 | alimdv 1835 | . . . 4 |
19 | dfss2 3056 | . . . 4 | |
20 | dfss2 3056 | . . . 4 | |
21 | 18, 19, 20 | 3imtr4g 204 | . . 3 |
22 | 21 | com12 30 | . 2 |
23 | 3, 22 | impbid2 142 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wal 1314 wceq 1316 wex 1453 wcel 1465 cvv 2660 wss 3041 cop 3500 cxp 4507 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-opab 3960 df-xp 4515 |
This theorem is referenced by: eqrelrel 4610 |
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