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Mirrors > Home > ILE Home > Th. List > ssrel | Unicode version |
Description: A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
ssrel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 2939 | . . 3 | |
2 | 1 | alrimivv 1755 | . 2 |
3 | eleq1 2100 | . . . . . . . . . . 11 | |
4 | eleq1 2100 | . . . . . . . . . . 11 | |
5 | 3, 4 | imbi12d 223 | . . . . . . . . . 10 |
6 | 5 | biimprcd 149 | . . . . . . . . 9 |
7 | 6 | 2alimi 1345 | . . . . . . . 8 |
8 | 19.23vv 1764 | . . . . . . . 8 | |
9 | 7, 8 | sylib 127 | . . . . . . 7 |
10 | 9 | com23 72 | . . . . . 6 |
11 | 10 | a2d 23 | . . . . 5 |
12 | 11 | alimdv 1759 | . . . 4 |
13 | df-rel 4352 | . . . . 5 | |
14 | dfss2 2934 | . . . . 5 | |
15 | elvv 4402 | . . . . . . 7 | |
16 | 15 | imbi2i 215 | . . . . . 6 |
17 | 16 | albii 1359 | . . . . 5 |
18 | 13, 14, 17 | 3bitri 195 | . . . 4 |
19 | dfss2 2934 | . . . 4 | |
20 | 12, 18, 19 | 3imtr4g 194 | . . 3 |
21 | 20 | com12 27 | . 2 |
22 | 2, 21 | 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 wrel 4350 |
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 df-rel 4352 |
This theorem is referenced by: eqrel 4429 relssi 4431 relssdv 4432 cotr 4706 cnvsym 4708 intasym 4709 intirr 4711 codir 4713 qfto 4714 |
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