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Mirrors > Home > ILE Home > Th. List > eliunxp | Unicode version |
Description: Membership in a union of cross products. Analogue of elxp 4602 for nonconstant . (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
eliunxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 4694 | . . . . . 6 | |
2 | 1 | rgenw 2512 | . . . . 5 |
3 | reliun 4706 | . . . . 5 | |
4 | 2, 3 | mpbir 145 | . . . 4 |
5 | elrel 4687 | . . . 4 | |
6 | 4, 5 | mpan 421 | . . 3 |
7 | 6 | pm4.71ri 390 | . 2 |
8 | nfiu1 3879 | . . . 4 | |
9 | 8 | nfel2 2312 | . . 3 |
10 | 9 | 19.41 1666 | . 2 |
11 | 19.41v 1882 | . . . 4 | |
12 | eleq1 2220 | . . . . . . 7 | |
13 | opeliunxp 4640 | . . . . . . 7 | |
14 | 12, 13 | bitrdi 195 | . . . . . 6 |
15 | 14 | pm5.32i 450 | . . . . 5 |
16 | 15 | exbii 1585 | . . . 4 |
17 | 11, 16 | bitr3i 185 | . . 3 |
18 | 17 | exbii 1585 | . 2 |
19 | 7, 10, 18 | 3bitr2i 207 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1335 wex 1472 wcel 2128 wral 2435 csn 3560 cop 3563 ciun 3849 cxp 4583 wrel 4590 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-iun 3851 df-opab 4026 df-xp 4591 df-rel 4592 |
This theorem is referenced by: raliunxp 4726 rexiunxp 4727 dfmpt3 5291 mpomptx 5909 fisumcom2 11330 fprodcom2fi 11518 |
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