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Mirrors > Home > ILE Home > Th. List > ssrelrel | Unicode version |
Description: A subclass relationship determined by ordered triples. Use relrelss 4864 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 2993 | . . . 4 | |
2 | 1 | alrimiv 1795 | . . 3 |
3 | 2 | alrimivv 1796 | . 2 |
4 | elvvv 4421 | . . . . . . . 8 | |
5 | eleq1 2141 | . . . . . . . . . . . . . 14 | |
6 | eleq1 2141 | . . . . . . . . . . . . . 14 | |
7 | 5, 6 | imbi12d 232 | . . . . . . . . . . . . 13 |
8 | 7 | biimprcd 158 | . . . . . . . . . . . 12 |
9 | 8 | alimi 1384 | . . . . . . . . . . 11 |
10 | 19.23v 1804 | . . . . . . . . . . 11 | |
11 | 9, 10 | sylib 120 | . . . . . . . . . 10 |
12 | 11 | 2alimi 1385 | . . . . . . . . 9 |
13 | 19.23vv 1805 | . . . . . . . . 9 | |
14 | 12, 13 | sylib 120 | . . . . . . . 8 |
15 | 4, 14 | syl5bi 150 | . . . . . . 7 |
16 | 15 | com23 77 | . . . . . 6 |
17 | 16 | a2d 26 | . . . . 5 |
18 | 17 | alimdv 1800 | . . . 4 |
19 | dfss2 2988 | . . . 4 | |
20 | dfss2 2988 | . . . 4 | |
21 | 18, 19, 20 | 3imtr4g 203 | . . 3 |
22 | 21 | com12 30 | . 2 |
23 | 3, 22 | impbid2 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 103 wal 1282 wceq 1284 wex 1421 wcel 1433 cvv 2601 wss 2973 cop 3401 cxp 4361 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-opab 3840 df-xp 4369 |
This theorem is referenced by: eqrelrel 4459 |
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