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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 5074. (Contributed by NM, 9-Oct-2004.) |
Ref | Expression |
---|---|
elxp6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2723 | . 2 | |
2 | opexg 4189 | . . . 4 | |
3 | 2 | adantl 275 | . . 3 |
4 | eleq1 2220 | . . . 4 | |
5 | 4 | adantr 274 | . . 3 |
6 | 3, 5 | mpbird 166 | . 2 |
7 | elxp4 5074 | . . 3 | |
8 | 1stvalg 6091 | . . . . . 6 | |
9 | 2ndvalg 6092 | . . . . . 6 | |
10 | 8, 9 | opeq12d 3750 | . . . . 5 |
11 | 10 | eqeq2d 2169 | . . . 4 |
12 | 8 | eleq1d 2226 | . . . . 5 |
13 | 9 | eleq1d 2226 | . . . . 5 |
14 | 12, 13 | anbi12d 465 | . . . 4 |
15 | 11, 14 | anbi12d 465 | . . 3 |
16 | 7, 15 | bitr4id 198 | . 2 |
17 | 1, 6, 16 | pm5.21nii 694 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1335 wcel 2128 cvv 2712 csn 3560 cop 3563 cuni 3773 cxp 4585 cdm 4587 crn 4588 cfv 5171 c1st 6087 c2nd 6088 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-iota 5136 df-fun 5173 df-fv 5179 df-1st 6089 df-2nd 6090 |
This theorem is referenced by: elxp7 6119 oprssdmm 6120 eqopi 6121 1st2nd2 6124 eldju2ndl 7017 eldju2ndr 7018 qredeu 11978 qnumdencl 12066 tx1cn 12711 tx2cn 12712 psmetxrge0 12774 xmetxpbl 12950 |
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