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Mirrors > Home > ILE Home > Th. List > ssrelrel | Unicode version |
Description: A subclass relationship determined by ordered triples. Use relrelss 4844 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 2939 | . . . 4 | |
2 | 1 | alrimiv 1754 | . . 3 |
3 | 2 | alrimivv 1755 | . 2 |
4 | elvvv 4403 | . . . . . . . 8 | |
5 | eleq1 2100 | . . . . . . . . . . . . . 14 | |
6 | eleq1 2100 | . . . . . . . . . . . . . 14 | |
7 | 5, 6 | imbi12d 223 | . . . . . . . . . . . . 13 |
8 | 7 | biimprcd 149 | . . . . . . . . . . . 12 |
9 | 8 | alimi 1344 | . . . . . . . . . . 11 |
10 | 19.23v 1763 | . . . . . . . . . . 11 | |
11 | 9, 10 | sylib 127 | . . . . . . . . . 10 |
12 | 11 | 2alimi 1345 | . . . . . . . . 9 |
13 | 19.23vv 1764 | . . . . . . . . 9 | |
14 | 12, 13 | sylib 127 | . . . . . . . 8 |
15 | 4, 14 | syl5bi 141 | . . . . . . 7 |
16 | 15 | com23 72 | . . . . . 6 |
17 | 16 | a2d 23 | . . . . 5 |
18 | 17 | alimdv 1759 | . . . 4 |
19 | dfss2 2934 | . . . 4 | |
20 | dfss2 2934 | . . . 4 | |
21 | 18, 19, 20 | 3imtr4g 194 | . . 3 |
22 | 21 | com12 27 | . 2 |
23 | 3, 22 | impbid2 131 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 98 wal 1241 wceq 1243 wex 1381 wcel 1393 cvv 2557 wss 2917 cop 3378 cxp 4343 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-xp 4351 |
This theorem is referenced by: eqrelrel 4441 |
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