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Mirrors > Home > ILE Home > Th. List > opeliunxp2 | Unicode version |
Description: Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.) |
Ref | Expression |
---|---|
opeliunxp2.1 |
Ref | Expression |
---|---|
opeliunxp2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3977 | . . 3 | |
2 | relxp 4707 | . . . . . 6 | |
3 | 2 | rgenw 2519 | . . . . 5 |
4 | reliun 4719 | . . . . 5 | |
5 | 3, 4 | mpbir 145 | . . . 4 |
6 | 5 | brrelex1i 4641 | . . 3 |
7 | 1, 6 | sylbir 134 | . 2 |
8 | elex 2732 | . . 3 | |
9 | 8 | adantr 274 | . 2 |
10 | nfcv 2306 | . . 3 | |
11 | nfiu1 3890 | . . . . 5 | |
12 | 11 | nfel2 2319 | . . . 4 |
13 | nfv 1515 | . . . 4 | |
14 | 12, 13 | nfbi 1576 | . . 3 |
15 | opeq1 3752 | . . . . 5 | |
16 | 15 | eleq1d 2233 | . . . 4 |
17 | eleq1 2227 | . . . . 5 | |
18 | opeliunxp2.1 | . . . . . 6 | |
19 | 18 | eleq2d 2234 | . . . . 5 |
20 | 17, 19 | anbi12d 465 | . . . 4 |
21 | 16, 20 | bibi12d 234 | . . 3 |
22 | opeliunxp 4653 | . . 3 | |
23 | 10, 14, 21, 22 | vtoclgf 2779 | . 2 |
24 | 7, 9, 23 | pm5.21nii 694 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wral 2442 cvv 2721 csn 3570 cop 3573 ciun 3860 class class class wbr 3976 cxp 4596 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-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-iun 3862 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 |
This theorem is referenced by: mpoxopn0yelv 6198 eldvap 13198 |
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