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Mirrors > Home > MPE Home > Th. List > ssrel2 | Structured version Visualization version Unicode version |
Description: A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 5207 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.) |
Ref | Expression |
---|---|
ssrel2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3597 | . . . 4 | |
2 | 1 | a1d 25 | . . 3 |
3 | 2 | ralrimivv 2970 | . 2 |
4 | eleq1 2689 | . . . . . . . . . . 11 | |
5 | eleq1 2689 | . . . . . . . . . . 11 | |
6 | 4, 5 | imbi12d 334 | . . . . . . . . . 10 |
7 | 6 | biimprcd 240 | . . . . . . . . 9 |
8 | 7 | 2ralimi 2953 | . . . . . . . 8 |
9 | r19.23v 3023 | . . . . . . . . . 10 | |
10 | 9 | ralbii 2980 | . . . . . . . . 9 |
11 | r19.23v 3023 | . . . . . . . . 9 | |
12 | 10, 11 | bitri 264 | . . . . . . . 8 |
13 | 8, 12 | sylib 208 | . . . . . . 7 |
14 | 13 | com23 86 | . . . . . 6 |
15 | 14 | a2d 29 | . . . . 5 |
16 | 15 | alimdv 1845 | . . . 4 |
17 | dfss2 3591 | . . . . 5 | |
18 | elxp2 5132 | . . . . . . 7 | |
19 | 18 | imbi2i 326 | . . . . . 6 |
20 | 19 | albii 1747 | . . . . 5 |
21 | 17, 20 | bitri 264 | . . . 4 |
22 | dfss2 3591 | . . . 4 | |
23 | 16, 21, 22 | 3imtr4g 285 | . . 3 |
24 | 23 | com12 32 | . 2 |
25 | 3, 24 | impbid2 216 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wal 1481 wceq 1483 wcel 1990 wral 2912 wrex 2913 wss 3574 cop 4183 cxp 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-xp 5120 |
This theorem is referenced by: metuel2 22370 isarchi 29736 |
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