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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 4855 |
. . . . 5
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2 | elrel 4649 |
. . . . 5
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3 | 1, 2 | mpan 421 |
. . . 4
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4 | eleq1 2203 |
. . . . . . . . 9
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5 | 4 | biimpd 143 |
. . . . . . . 8
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6 | vex 2692 |
. . . . . . . . . . 11
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7 | 6 | opelres 4832 |
. . . . . . . . . 10
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8 | 7 | biimpi 119 |
. . . . . . . . 9
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9 | 8 | ancomd 265 |
. . . . . . . 8
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10 | 5, 9 | syl6com 35 |
. . . . . . 7
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11 | 10 | ancld 323 |
. . . . . 6
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12 | an12 551 |
. . . . . 6
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13 | 11, 12 | syl6ib 160 |
. . . . 5
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14 | 13 | 2eximdv 1855 |
. . . 4
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15 | 3, 14 | mpd 13 |
. . 3
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16 | rexcom4 2712 |
. . . 4
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17 | df-rex 2423 |
. . . . 5
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18 | 17 | exbii 1585 |
. . . 4
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19 | excom 1643 |
. . . 4
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20 | 16, 18, 19 | 3bitri 205 |
. . 3
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21 | 15, 20 | sylibr 133 |
. 2
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22 | 7 | simplbi2com 1421 |
. . . . . 6
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23 | 4 | biimprd 157 |
. . . . . 6
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24 | 22, 23 | syl9 72 |
. . . . 5
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25 | 24 | impd 252 |
. . . 4
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26 | 25 | exlimdv 1792 |
. . 3
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27 | 26 | rexlimiv 2546 |
. 2
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28 | 21, 27 | impbii 125 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-opab 3998 df-xp 4553 df-rel 4554 df-res 4559 |
This theorem is referenced by: elsnres 4864 |
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