Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > eliunxp | Unicode version |
Description: Membership in a union of cross products. Analogue of elxp 4556 for nonconstant . (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
eliunxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 4648 | . . . . . 6 | |
2 | 1 | rgenw 2487 | . . . . 5 |
3 | reliun 4660 | . . . . 5 | |
4 | 2, 3 | mpbir 145 | . . . 4 |
5 | elrel 4641 | . . . 4 | |
6 | 4, 5 | mpan 420 | . . 3 |
7 | 6 | pm4.71ri 389 | . 2 |
8 | nfiu1 3843 | . . . 4 | |
9 | 8 | nfel2 2294 | . . 3 |
10 | 9 | 19.41 1664 | . 2 |
11 | 19.41v 1874 | . . . 4 | |
12 | eleq1 2202 | . . . . . . 7 | |
13 | opeliunxp 4594 | . . . . . . 7 | |
14 | 12, 13 | syl6bb 195 | . . . . . 6 |
15 | 14 | pm5.32i 449 | . . . . 5 |
16 | 15 | exbii 1584 | . . . 4 |
17 | 11, 16 | bitr3i 185 | . . 3 |
18 | 17 | exbii 1584 | . 2 |
19 | 7, 10, 18 | 3bitr2i 207 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wral 2416 csn 3527 cop 3530 ciun 3813 cxp 4537 wrel 4544 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-iun 3815 df-opab 3990 df-xp 4545 df-rel 4546 |
This theorem is referenced by: raliunxp 4680 rexiunxp 4681 dfmpt3 5245 mpomptx 5862 fisumcom2 11207 |
Copyright terms: Public domain | W3C validator |