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Mirrors > Home > ILE Home > Th. List > eliunxp | Unicode version |
Description: Membership in a union of cross products. Analogue of elxp 4380 for nonconstant . (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
eliunxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 4465 | . . . . . 6 | |
2 | 1 | rgenw 2418 | . . . . 5 |
3 | reliun 4476 | . . . . 5 | |
4 | 2, 3 | mpbir 144 | . . . 4 |
5 | elrel 4460 | . . . 4 | |
6 | 4, 5 | mpan 414 | . . 3 |
7 | 6 | pm4.71ri 384 | . 2 |
8 | nfiu1 3708 | . . . 4 | |
9 | 8 | nfel2 2231 | . . 3 |
10 | 9 | 19.41 1616 | . 2 |
11 | 19.41v 1823 | . . . 4 | |
12 | eleq1 2141 | . . . . . . 7 | |
13 | opeliunxp 4413 | . . . . . . 7 | |
14 | 12, 13 | syl6bb 194 | . . . . . 6 |
15 | 14 | pm5.32i 441 | . . . . 5 |
16 | 15 | exbii 1536 | . . . 4 |
17 | 11, 16 | bitr3i 184 | . . 3 |
18 | 17 | exbii 1536 | . 2 |
19 | 7, 10, 18 | 3bitr2i 206 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 wral 2348 csn 3398 cop 3401 ciun 3678 cxp 4361 wrel 4368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-iun 3680 df-opab 3840 df-xp 4369 df-rel 4370 |
This theorem is referenced by: raliunxp 4495 rexiunxp 4496 dfmpt3 5041 mpt2mptx 5615 |
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