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Mirrors > Home > ILE Home > Th. List > opeliunxp | Unicode version |
Description: Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.) |
Ref | Expression |
---|---|
opeliunxp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2763 |
. 2
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2 | opexg 4243 |
. 2
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3 | df-rex 2474 |
. . . . . 6
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4 | nfv 1539 |
. . . . . . 7
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5 | nfs1v 1951 |
. . . . . . . 8
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6 | nfcv 2332 |
. . . . . . . . . 10
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7 | nfcsb1v 3105 |
. . . . . . . . . 10
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8 | 6, 7 | nfxp 4668 |
. . . . . . . . 9
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9 | 8 | nfcri 2326 |
. . . . . . . 8
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10 | 5, 9 | nfan 1576 |
. . . . . . 7
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11 | sbequ12 1782 |
. . . . . . . 8
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12 | sneq 3618 |
. . . . . . . . . 10
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13 | csbeq1a 3081 |
. . . . . . . . . 10
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14 | 12, 13 | xpeq12d 4666 |
. . . . . . . . 9
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15 | 14 | eleq2d 2259 |
. . . . . . . 8
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16 | 11, 15 | anbi12d 473 |
. . . . . . 7
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17 | 4, 10, 16 | cbvex 1767 |
. . . . . 6
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18 | 3, 17 | bitri 184 |
. . . . 5
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19 | eleq1 2252 |
. . . . . . 7
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20 | 19 | anbi2d 464 |
. . . . . 6
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21 | 20 | exbidv 1836 |
. . . . 5
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22 | 18, 21 | bitrid 192 |
. . . 4
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23 | df-iun 3903 |
. . . 4
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24 | 22, 23 | elab2g 2899 |
. . 3
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25 | opelxp 4671 |
. . . . . . 7
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26 | 25 | anbi2i 457 |
. . . . . 6
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27 | an12 561 |
. . . . . 6
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28 | velsn 3624 |
. . . . . . . 8
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29 | equcom 1717 |
. . . . . . . 8
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30 | 28, 29 | bitri 184 |
. . . . . . 7
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31 | 30 | anbi1i 458 |
. . . . . 6
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32 | 26, 27, 31 | 3bitri 206 |
. . . . 5
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33 | 32 | exbii 1616 |
. . . 4
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34 | vex 2755 |
. . . . 5
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35 | sbequ12r 1783 |
. . . . . 6
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36 | 13 | equcoms 1719 |
. . . . . . . 8
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37 | 36 | eqcomd 2195 |
. . . . . . 7
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38 | 37 | eleq2d 2259 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 35, 38 | anbi12d 473 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 34, 39 | ceqsexv 2791 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 33, 40 | bitri 184 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | 24, 41 | bitrdi 196 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | 1, 2, 42 | pm5.21nii 705 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-iun 3903 df-opab 4080 df-xp 4647 |
This theorem is referenced by: eliunxp 4781 opeliunxp2 4782 opeliunxp2f 6257 |
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