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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 3086 | . . 3 | |
2 | 1 | alrimivv 1847 | . 2 |
3 | eleq1 2200 | . . . . . . . . . . 11 | |
4 | eleq1 2200 | . . . . . . . . . . 11 | |
5 | 3, 4 | imbi12d 233 | . . . . . . . . . 10 |
6 | 5 | biimprcd 159 | . . . . . . . . 9 |
7 | 6 | 2alimi 1432 | . . . . . . . 8 |
8 | 19.23vv 1856 | . . . . . . . 8 | |
9 | 7, 8 | sylib 121 | . . . . . . 7 |
10 | 9 | com23 78 | . . . . . 6 |
11 | 10 | a2d 26 | . . . . 5 |
12 | 11 | alimdv 1851 | . . . 4 |
13 | df-rel 4541 | . . . . 5 | |
14 | dfss2 3081 | . . . . 5 | |
15 | elvv 4596 | . . . . . . 7 | |
16 | 15 | imbi2i 225 | . . . . . 6 |
17 | 16 | albii 1446 | . . . . 5 |
18 | 13, 14, 17 | 3bitri 205 | . . . 4 |
19 | dfss2 3081 | . . . 4 | |
20 | 12, 18, 19 | 3imtr4g 204 | . . 3 |
21 | 20 | com12 30 | . 2 |
22 | 2, 21 | impbid2 142 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wal 1329 wceq 1331 wex 1468 wcel 1480 cvv 2681 wss 3066 cop 3525 cxp 4532 wrel 4539 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 df-xp 4540 df-rel 4541 |
This theorem is referenced by: eqrel 4623 relssi 4625 relssdv 4626 cotr 4915 cnvsym 4917 intasym 4918 intirr 4920 codir 4922 qfto 4923 |
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