Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > elxp6 | Unicode version |
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4996. (Contributed by NM, 9-Oct-2004.) |
Ref | Expression |
---|---|
elxp6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2671 | . 2 | |
2 | opexg 4120 | . . . 4 | |
3 | 2 | adantl 275 | . . 3 |
4 | eleq1 2180 | . . . 4 | |
5 | 4 | adantr 274 | . . 3 |
6 | 3, 5 | mpbird 166 | . 2 |
7 | 1stvalg 6008 | . . . . . 6 | |
8 | 2ndvalg 6009 | . . . . . 6 | |
9 | 7, 8 | opeq12d 3683 | . . . . 5 |
10 | 9 | eqeq2d 2129 | . . . 4 |
11 | 7 | eleq1d 2186 | . . . . 5 |
12 | 8 | eleq1d 2186 | . . . . 5 |
13 | 11, 12 | anbi12d 464 | . . . 4 |
14 | 10, 13 | anbi12d 464 | . . 3 |
15 | elxp4 4996 | . . 3 | |
16 | 14, 15 | syl6rbbr 198 | . 2 |
17 | 1, 6, 16 | pm5.21nii 678 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1316 wcel 1465 cvv 2660 csn 3497 cop 3500 cuni 3706 cxp 4507 cdm 4509 crn 4510 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-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-fv 5101 df-1st 6006 df-2nd 6007 |
This theorem is referenced by: elxp7 6036 oprssdmm 6037 eqopi 6038 1st2nd2 6041 eldju2ndl 6925 eldju2ndr 6926 qredeu 11705 qnumdencl 11792 tx1cn 12365 tx2cn 12366 psmetxrge0 12428 xmetxpbl 12604 |
Copyright terms: Public domain | W3C validator |