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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 4627 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.) |
Ref | Expression |
---|---|
ssrel2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3091 | . . . 4 | |
2 | 1 | a1d 22 | . . 3 |
3 | 2 | ralrimivv 2513 | . 2 |
4 | eleq1 2202 | . . . . . . . . . . . 12 | |
5 | eleq1 2202 | . . . . . . . . . . . 12 | |
6 | 4, 5 | imbi12d 233 | . . . . . . . . . . 11 |
7 | 6 | biimprcd 159 | . . . . . . . . . 10 |
8 | 7 | ralimi 2495 | . . . . . . . . 9 |
9 | 8 | ralimi 2495 | . . . . . . . 8 |
10 | r19.23v 2541 | . . . . . . . . . 10 | |
11 | 10 | ralbii 2441 | . . . . . . . . 9 |
12 | r19.23v 2541 | . . . . . . . . 9 | |
13 | 11, 12 | bitri 183 | . . . . . . . 8 |
14 | 9, 13 | sylib 121 | . . . . . . 7 |
15 | 14 | com23 78 | . . . . . 6 |
16 | 15 | a2d 26 | . . . . 5 |
17 | 16 | alimdv 1851 | . . . 4 |
18 | dfss2 3086 | . . . . 5 | |
19 | elxp2 4557 | . . . . . . 7 | |
20 | 19 | imbi2i 225 | . . . . . 6 |
21 | 20 | albii 1446 | . . . . 5 |
22 | 18, 21 | bitri 183 | . . . 4 |
23 | dfss2 3086 | . . . 4 | |
24 | 17, 22, 23 | 3imtr4g 204 | . . 3 |
25 | 24 | com12 30 | . 2 |
26 | 3, 25 | impbid2 142 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wceq 1331 wcel 1480 wral 2416 wrex 2417 wss 3071 cop 3530 cxp 4537 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-xp 4545 |
This theorem is referenced by: (None) |
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