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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 2644 |
. 2
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2 | opexg 4079 |
. 2
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3 | df-rex 2376 |
. . . . . 6
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4 | nfv 1473 |
. . . . . . 7
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5 | nfs1v 1870 |
. . . . . . . 8
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6 | nfcv 2235 |
. . . . . . . . . 10
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7 | nfcsb1v 2977 |
. . . . . . . . . 10
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8 | 6, 7 | nfxp 4494 |
. . . . . . . . 9
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9 | 8 | nfcri 2229 |
. . . . . . . 8
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10 | 5, 9 | nfan 1509 |
. . . . . . 7
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11 | sbequ12 1708 |
. . . . . . . 8
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12 | sneq 3477 |
. . . . . . . . . 10
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13 | csbeq1a 2955 |
. . . . . . . . . 10
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14 | 12, 13 | xpeq12d 4492 |
. . . . . . . . 9
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15 | 14 | eleq2d 2164 |
. . . . . . . 8
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16 | 11, 15 | anbi12d 458 |
. . . . . . 7
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17 | 4, 10, 16 | cbvex 1693 |
. . . . . 6
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18 | 3, 17 | bitri 183 |
. . . . 5
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19 | eleq1 2157 |
. . . . . . 7
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20 | 19 | anbi2d 453 |
. . . . . 6
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21 | 20 | exbidv 1760 |
. . . . 5
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22 | 18, 21 | syl5bb 191 |
. . . 4
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23 | df-iun 3754 |
. . . 4
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24 | 22, 23 | elab2g 2776 |
. . 3
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25 | opelxp 4497 |
. . . . . . 7
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26 | 25 | anbi2i 446 |
. . . . . 6
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27 | an12 529 |
. . . . . 6
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28 | velsn 3483 |
. . . . . . . 8
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29 | equcom 1646 |
. . . . . . . 8
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30 | 28, 29 | bitri 183 |
. . . . . . 7
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31 | 30 | anbi1i 447 |
. . . . . 6
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32 | 26, 27, 31 | 3bitri 205 |
. . . . 5
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33 | 32 | exbii 1548 |
. . . 4
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34 | vex 2636 |
. . . . 5
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35 | sbequ12r 1709 |
. . . . . 6
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36 | 13 | equcoms 1648 |
. . . . . . . 8
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37 | 36 | eqcomd 2100 |
. . . . . . 7
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38 | 37 | eleq2d 2164 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 35, 38 | anbi12d 458 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 34, 39 | ceqsexv 2672 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 33, 40 | bitri 183 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | 24, 41 | syl6bb 195 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | 1, 2, 42 | pm5.21nii 658 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-sbc 2855 df-csb 2948 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-iun 3754 df-opab 3922 df-xp 4473 |
This theorem is referenced by: eliunxp 4606 opeliunxp2 4607 opeliunxp2f 6041 |
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