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Mirrors > Home > ILE Home > Th. List > elres | Unicode version |
Description: Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.) |
Ref | Expression |
---|---|
elres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 4639 | . . . . 5 | |
2 | elrel 4442 | . . . . 5 | |
3 | 1, 2 | mpan 400 | . . . 4 |
4 | eleq1 2100 | . . . . . . . . 9 | |
5 | 4 | biimpd 132 | . . . . . . . 8 |
6 | vex 2560 | . . . . . . . . . . 11 | |
7 | 6 | opelres 4617 | . . . . . . . . . 10 |
8 | 7 | biimpi 113 | . . . . . . . . 9 |
9 | 8 | ancomd 254 | . . . . . . . 8 |
10 | 5, 9 | syl6com 31 | . . . . . . 7 |
11 | 10 | ancld 308 | . . . . . 6 |
12 | an12 495 | . . . . . 6 | |
13 | 11, 12 | syl6ib 150 | . . . . 5 |
14 | 13 | 2eximdv 1762 | . . . 4 |
15 | 3, 14 | mpd 13 | . . 3 |
16 | rexcom4 2577 | . . . 4 | |
17 | df-rex 2312 | . . . . 5 | |
18 | 17 | exbii 1496 | . . . 4 |
19 | excom 1554 | . . . 4 | |
20 | 16, 18, 19 | 3bitri 195 | . . 3 |
21 | 15, 20 | sylibr 137 | . 2 |
22 | 7 | simplbi2com 1333 | . . . . . 6 |
23 | 4 | biimprd 147 | . . . . . 6 |
24 | 22, 23 | syl9 66 | . . . . 5 |
25 | 24 | impd 242 | . . . 4 |
26 | 25 | exlimdv 1700 | . . 3 |
27 | 26 | rexlimiv 2427 | . 2 |
28 | 21, 27 | impbii 117 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 wrex 2307 cop 3378 cres 4347 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-ral 2311 df-rex 2312 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 df-res 4357 |
This theorem is referenced by: elsnres 4647 |
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