Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ssrel2 | Unicode version |
Description: A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4446 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.) |
Ref | Expression |
---|---|
ssrel2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 2993 | . . . 4 | |
2 | 1 | a1d 22 | . . 3 |
3 | 2 | ralrimivv 2442 | . 2 |
4 | eleq1 2141 | . . . . . . . . . . . 12 | |
5 | eleq1 2141 | . . . . . . . . . . . 12 | |
6 | 4, 5 | imbi12d 232 | . . . . . . . . . . 11 |
7 | 6 | biimprcd 158 | . . . . . . . . . 10 |
8 | 7 | ralimi 2426 | . . . . . . . . 9 |
9 | 8 | ralimi 2426 | . . . . . . . 8 |
10 | r19.23v 2469 | . . . . . . . . . 10 | |
11 | 10 | ralbii 2372 | . . . . . . . . 9 |
12 | r19.23v 2469 | . . . . . . . . 9 | |
13 | 11, 12 | bitri 182 | . . . . . . . 8 |
14 | 9, 13 | sylib 120 | . . . . . . 7 |
15 | 14 | com23 77 | . . . . . 6 |
16 | 15 | a2d 26 | . . . . 5 |
17 | 16 | alimdv 1800 | . . . 4 |
18 | dfss2 2988 | . . . . 5 | |
19 | elxp2 4381 | . . . . . . 7 | |
20 | 19 | imbi2i 224 | . . . . . 6 |
21 | 20 | albii 1399 | . . . . 5 |
22 | 18, 21 | bitri 182 | . . . 4 |
23 | dfss2 2988 | . . . 4 | |
24 | 17, 22, 23 | 3imtr4g 203 | . . 3 |
25 | 24 | com12 30 | . 2 |
26 | 3, 25 | impbid2 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wal 1282 wceq 1284 wcel 1433 wral 2348 wrex 2349 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-ral 2353 df-rex 2354 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: (None) |
Copyright terms: Public domain | W3C validator |