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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 4906 | . . . . 5 | |
2 | elrel 4700 | . . . . 5 | |
3 | 1, 2 | mpan 421 | . . . 4 |
4 | eleq1 2227 | . . . . . . . . 9 | |
5 | 4 | biimpd 143 | . . . . . . . 8 |
6 | vex 2724 | . . . . . . . . . . 11 | |
7 | 6 | opelres 4883 | . . . . . . . . . 10 |
8 | 7 | biimpi 119 | . . . . . . . . 9 |
9 | 8 | ancomd 265 | . . . . . . . 8 |
10 | 5, 9 | syl6com 35 | . . . . . . 7 |
11 | 10 | ancld 323 | . . . . . 6 |
12 | an12 551 | . . . . . 6 | |
13 | 11, 12 | syl6ib 160 | . . . . 5 |
14 | 13 | 2eximdv 1869 | . . . 4 |
15 | 3, 14 | mpd 13 | . . 3 |
16 | rexcom4 2744 | . . . 4 | |
17 | df-rex 2448 | . . . . 5 | |
18 | 17 | exbii 1592 | . . . 4 |
19 | excom 1651 | . . . 4 | |
20 | 16, 18, 19 | 3bitri 205 | . . 3 |
21 | 15, 20 | sylibr 133 | . 2 |
22 | 7 | simplbi2com 1431 | . . . . . 6 |
23 | 4 | biimprd 157 | . . . . . 6 |
24 | 22, 23 | syl9 72 | . . . . 5 |
25 | 24 | impd 252 | . . . 4 |
26 | 25 | exlimdv 1806 | . . 3 |
27 | 26 | rexlimiv 2575 | . 2 |
28 | 21, 27 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 wrex 2443 cop 3573 cres 4600 wrel 4603 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-opab 4038 df-xp 4604 df-rel 4605 df-res 4610 |
This theorem is referenced by: elsnres 4915 |
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