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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 4657 | . . . . 5 | |
2 | elrel 4460 | . . . . 5 | |
3 | 1, 2 | mpan 414 | . . . 4 |
4 | eleq1 2141 | . . . . . . . . 9 | |
5 | 4 | biimpd 142 | . . . . . . . 8 |
6 | vex 2604 | . . . . . . . . . . 11 | |
7 | 6 | opelres 4635 | . . . . . . . . . 10 |
8 | 7 | biimpi 118 | . . . . . . . . 9 |
9 | 8 | ancomd 263 | . . . . . . . 8 |
10 | 5, 9 | syl6com 35 | . . . . . . 7 |
11 | 10 | ancld 318 | . . . . . 6 |
12 | an12 525 | . . . . . 6 | |
13 | 11, 12 | syl6ib 159 | . . . . 5 |
14 | 13 | 2eximdv 1803 | . . . 4 |
15 | 3, 14 | mpd 13 | . . 3 |
16 | rexcom4 2622 | . . . 4 | |
17 | df-rex 2354 | . . . . 5 | |
18 | 17 | exbii 1536 | . . . 4 |
19 | excom 1594 | . . . 4 | |
20 | 16, 18, 19 | 3bitri 204 | . . 3 |
21 | 15, 20 | sylibr 132 | . 2 |
22 | 7 | simplbi2com 1373 | . . . . . 6 |
23 | 4 | biimprd 156 | . . . . . 6 |
24 | 22, 23 | syl9 71 | . . . . 5 |
25 | 24 | impd 251 | . . . 4 |
26 | 25 | exlimdv 1740 | . . 3 |
27 | 26 | rexlimiv 2471 | . 2 |
28 | 21, 27 | impbii 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 wrex 2349 cop 3401 cres 4365 wrel 4368 |
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 df-rel 4370 df-res 4375 |
This theorem is referenced by: elsnres 4665 |
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