Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > unielxp | Unicode version |
Description: The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.) |
Ref | Expression |
---|---|
unielxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp7 6036 | . 2 | |
2 | elvvuni 4573 | . . . 4 | |
3 | 2 | adantr 274 | . . 3 |
4 | simprl 505 | . . . . . 6 | |
5 | eleq2 2181 | . . . . . . . 8 | |
6 | eleq1 2180 | . . . . . . . . 9 | |
7 | fveq2 5389 | . . . . . . . . . . 11 | |
8 | 7 | eleq1d 2186 | . . . . . . . . . 10 |
9 | fveq2 5389 | . . . . . . . . . . 11 | |
10 | 9 | eleq1d 2186 | . . . . . . . . . 10 |
11 | 8, 10 | anbi12d 464 | . . . . . . . . 9 |
12 | 6, 11 | anbi12d 464 | . . . . . . . 8 |
13 | 5, 12 | anbi12d 464 | . . . . . . 7 |
14 | 13 | spcegv 2748 | . . . . . 6 |
15 | 4, 14 | mpcom 36 | . . . . 5 |
16 | eluniab 3718 | . . . . 5 | |
17 | 15, 16 | sylibr 133 | . . . 4 |
18 | xp2 6039 | . . . . . 6 | |
19 | df-rab 2402 | . . . . . 6 | |
20 | 18, 19 | eqtri 2138 | . . . . 5 |
21 | 20 | unieqi 3716 | . . . 4 |
22 | 17, 21 | eleqtrrdi 2211 | . . 3 |
23 | 3, 22 | mpancom 418 | . 2 |
24 | 1, 23 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wex 1453 wcel 1465 cab 2103 crab 2397 cvv 2660 cuni 3706 cxp 4507 cfv 5093 c1st 6004 c2nd 6005 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fo 5099 df-fv 5101 df-1st 6006 df-2nd 6007 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |