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Mirrors > Home > ILE Home > Th. List > eliunxp | Unicode version |
Description: Membership in a union of cross products. Analogue of elxp 4628 for nonconstant . (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
eliunxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 4720 | . . . . . 6 | |
2 | 1 | rgenw 2525 | . . . . 5 |
3 | reliun 4732 | . . . . 5 | |
4 | 2, 3 | mpbir 145 | . . . 4 |
5 | elrel 4713 | . . . 4 | |
6 | 4, 5 | mpan 422 | . . 3 |
7 | 6 | pm4.71ri 390 | . 2 |
8 | nfiu1 3903 | . . . 4 | |
9 | 8 | nfel2 2325 | . . 3 |
10 | 9 | 19.41 1679 | . 2 |
11 | 19.41v 1895 | . . . 4 | |
12 | eleq1 2233 | . . . . . . 7 | |
13 | opeliunxp 4666 | . . . . . . 7 | |
14 | 12, 13 | bitrdi 195 | . . . . . 6 |
15 | 14 | pm5.32i 451 | . . . . 5 |
16 | 15 | exbii 1598 | . . . 4 |
17 | 11, 16 | bitr3i 185 | . . 3 |
18 | 17 | exbii 1598 | . 2 |
19 | 7, 10, 18 | 3bitr2i 207 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1348 wex 1485 wcel 2141 wral 2448 csn 3583 cop 3586 ciun 3873 cxp 4609 wrel 4616 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-iun 3875 df-opab 4051 df-xp 4617 df-rel 4618 |
This theorem is referenced by: raliunxp 4752 rexiunxp 4753 dfmpt3 5320 mpomptx 5944 fisumcom2 11401 fprodcom2fi 11589 |
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