Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > elxp7 | Unicode version |
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5096. (Contributed by NM, 19-Aug-2006.) |
Ref | Expression |
---|---|
elxp7 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2741 | . 2 | |
2 | elex 2741 | . . 3 | |
3 | 2 | adantr 274 | . 2 |
4 | elxp6 6145 | . . 3 | |
5 | elxp6 6145 | . . . . 5 | |
6 | 1stexg 6143 | . . . . . . 7 | |
7 | 2ndexg 6144 | . . . . . . 7 | |
8 | 6, 7 | jca 304 | . . . . . 6 |
9 | 8 | biantrud 302 | . . . . 5 |
10 | 5, 9 | bitr4id 198 | . . . 4 |
11 | 10 | anbi1d 462 | . . 3 |
12 | 4, 11 | bitr4id 198 | . 2 |
13 | 1, 3, 12 | pm5.21nii 699 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1348 wcel 2141 cvv 2730 cop 3584 cxp 4607 cfv 5196 c1st 6114 c2nd 6115 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fo 5202 df-fv 5204 df-1st 6116 df-2nd 6117 |
This theorem is referenced by: xp2 6149 unielxp 6150 1stconst 6197 2ndconst 6198 f1od2 6211 |
Copyright terms: Public domain | W3C validator |