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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2671 | . 2 | |
2 | opexg 4120 | . 2 | |
3 | df-rex 2399 | . . . . . 6 | |
4 | nfv 1493 | . . . . . . 7 | |
5 | nfs1v 1892 | . . . . . . . 8 | |
6 | nfcv 2258 | . . . . . . . . . 10 | |
7 | nfcsb1v 3005 | . . . . . . . . . 10 | |
8 | 6, 7 | nfxp 4536 | . . . . . . . . 9 |
9 | 8 | nfcri 2252 | . . . . . . . 8 |
10 | 5, 9 | nfan 1529 | . . . . . . 7 |
11 | sbequ12 1729 | . . . . . . . 8 | |
12 | sneq 3508 | . . . . . . . . . 10 | |
13 | csbeq1a 2983 | . . . . . . . . . 10 | |
14 | 12, 13 | xpeq12d 4534 | . . . . . . . . 9 |
15 | 14 | eleq2d 2187 | . . . . . . . 8 |
16 | 11, 15 | anbi12d 464 | . . . . . . 7 |
17 | 4, 10, 16 | cbvex 1714 | . . . . . 6 |
18 | 3, 17 | bitri 183 | . . . . 5 |
19 | eleq1 2180 | . . . . . . 7 | |
20 | 19 | anbi2d 459 | . . . . . 6 |
21 | 20 | exbidv 1781 | . . . . 5 |
22 | 18, 21 | syl5bb 191 | . . . 4 |
23 | df-iun 3785 | . . . 4 | |
24 | 22, 23 | elab2g 2804 | . . 3 |
25 | opelxp 4539 | . . . . . . 7 | |
26 | 25 | anbi2i 452 | . . . . . 6 |
27 | an12 535 | . . . . . 6 | |
28 | velsn 3514 | . . . . . . . 8 | |
29 | equcom 1667 | . . . . . . . 8 | |
30 | 28, 29 | bitri 183 | . . . . . . 7 |
31 | 30 | anbi1i 453 | . . . . . 6 |
32 | 26, 27, 31 | 3bitri 205 | . . . . 5 |
33 | 32 | exbii 1569 | . . . 4 |
34 | vex 2663 | . . . . 5 | |
35 | sbequ12r 1730 | . . . . . 6 | |
36 | 13 | equcoms 1669 | . . . . . . . 8 |
37 | 36 | eqcomd 2123 | . . . . . . 7 |
38 | 37 | eleq2d 2187 | . . . . . 6 |
39 | 35, 38 | anbi12d 464 | . . . . 5 |
40 | 34, 39 | ceqsexv 2699 | . . . 4 |
41 | 33, 40 | bitri 183 | . . 3 |
42 | 24, 41 | syl6bb 195 | . 2 |
43 | 1, 2, 42 | pm5.21nii 678 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 wsb 1720 wrex 2394 cvv 2660 csb 2975 csn 3497 cop 3500 ciun 3783 cxp 4507 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-iun 3785 df-opab 3960 df-xp 4515 |
This theorem is referenced by: eliunxp 4648 opeliunxp2 4649 opeliunxp2f 6103 |
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