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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 3925 | . . 3 | |
2 | relxp 4643 | . . . . . 6 | |
3 | 2 | rgenw 2485 | . . . . 5 |
4 | reliun 4655 | . . . . 5 | |
5 | 3, 4 | mpbir 145 | . . . 4 |
6 | 5 | brrelex1i 4577 | . . 3 |
7 | 1, 6 | sylbir 134 | . 2 |
8 | elex 2692 | . . 3 | |
9 | 8 | adantr 274 | . 2 |
10 | nfcv 2279 | . . 3 | |
11 | nfiu1 3838 | . . . . 5 | |
12 | 11 | nfel2 2292 | . . . 4 |
13 | nfv 1508 | . . . 4 | |
14 | 12, 13 | nfbi 1568 | . . 3 |
15 | opeq1 3700 | . . . . 5 | |
16 | 15 | eleq1d 2206 | . . . 4 |
17 | eleq1 2200 | . . . . 5 | |
18 | opeliunxp2.1 | . . . . . 6 | |
19 | 18 | eleq2d 2207 | . . . . 5 |
20 | 17, 19 | anbi12d 464 | . . . 4 |
21 | 16, 20 | bibi12d 234 | . . 3 |
22 | opeliunxp 4589 | . . 3 | |
23 | 10, 14, 21, 22 | vtoclgf 2739 | . 2 |
24 | 7, 9, 23 | pm5.21nii 693 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2414 cvv 2681 csn 3522 cop 3525 ciun 3808 class class class wbr 3924 cxp 4532 wrel 4539 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-iun 3810 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 |
This theorem is referenced by: mpoxopn0yelv 6129 eldvap 12809 |
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