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Mirrors > Home > ILE Home > Th. List > elxp4 | Unicode version |
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 5022. (Contributed by NM, 17-Feb-2004.) |
Ref | Expression |
---|---|
elxp4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2692 | . 2 | |
2 | elex 2692 | . . . 4 | |
3 | elex 2692 | . . . 4 | |
4 | 2, 3 | anim12i 336 | . . 3 |
5 | opexg 4145 | . . . . 5 | |
6 | 5 | adantl 275 | . . . 4 |
7 | eleq1 2200 | . . . . 5 | |
8 | 7 | adantr 274 | . . . 4 |
9 | 6, 8 | mpbird 166 | . . 3 |
10 | 4, 9 | sylan2 284 | . 2 |
11 | elxp 4551 | . . . 4 | |
12 | 11 | a1i 9 | . . 3 |
13 | sneq 3533 | . . . . . . . . . . . . 13 | |
14 | 13 | rneqd 4763 | . . . . . . . . . . . 12 |
15 | 14 | unieqd 3742 | . . . . . . . . . . 11 |
16 | vex 2684 | . . . . . . . . . . . 12 | |
17 | vex 2684 | . . . . . . . . . . . 12 | |
18 | 16, 17 | op2nda 5018 | . . . . . . . . . . 11 |
19 | 15, 18 | syl6req 2187 | . . . . . . . . . 10 |
20 | 19 | pm4.71ri 389 | . . . . . . . . 9 |
21 | 20 | anbi1i 453 | . . . . . . . 8 |
22 | anass 398 | . . . . . . . 8 | |
23 | 21, 22 | bitri 183 | . . . . . . 7 |
24 | 23 | exbii 1584 | . . . . . 6 |
25 | snexg 4103 | . . . . . . . . 9 | |
26 | rnexg 4799 | . . . . . . . . 9 | |
27 | 25, 26 | syl 14 | . . . . . . . 8 |
28 | uniexg 4356 | . . . . . . . 8 | |
29 | 27, 28 | syl 14 | . . . . . . 7 |
30 | opeq2 3701 | . . . . . . . . . 10 | |
31 | 30 | eqeq2d 2149 | . . . . . . . . 9 |
32 | eleq1 2200 | . . . . . . . . . 10 | |
33 | 32 | anbi2d 459 | . . . . . . . . 9 |
34 | 31, 33 | anbi12d 464 | . . . . . . . 8 |
35 | 34 | ceqsexgv 2809 | . . . . . . 7 |
36 | 29, 35 | syl 14 | . . . . . 6 |
37 | 24, 36 | syl5bb 191 | . . . . 5 |
38 | sneq 3533 | . . . . . . . . . . . 12 | |
39 | 38 | dmeqd 4736 | . . . . . . . . . . 11 |
40 | 39 | unieqd 3742 | . . . . . . . . . 10 |
41 | 40 | adantl 275 | . . . . . . . . 9 |
42 | dmsnopg 5005 | . . . . . . . . . . . . 13 | |
43 | 29, 42 | syl 14 | . . . . . . . . . . . 12 |
44 | 43 | unieqd 3742 | . . . . . . . . . . 11 |
45 | 16 | unisn 3747 | . . . . . . . . . . 11 |
46 | 44, 45 | syl6eq 2186 | . . . . . . . . . 10 |
47 | 46 | adantr 274 | . . . . . . . . 9 |
48 | 41, 47 | eqtr2d 2171 | . . . . . . . 8 |
49 | 48 | ex 114 | . . . . . . 7 |
50 | 49 | pm4.71rd 391 | . . . . . 6 |
51 | 50 | anbi1d 460 | . . . . 5 |
52 | anass 398 | . . . . . 6 | |
53 | 52 | a1i 9 | . . . . 5 |
54 | 37, 51, 53 | 3bitrd 213 | . . . 4 |
55 | 54 | exbidv 1797 | . . 3 |
56 | dmexg 4798 | . . . . . 6 | |
57 | 25, 56 | syl 14 | . . . . 5 |
58 | uniexg 4356 | . . . . 5 | |
59 | 57, 58 | syl 14 | . . . 4 |
60 | opeq1 3700 | . . . . . . 7 | |
61 | 60 | eqeq2d 2149 | . . . . . 6 |
62 | eleq1 2200 | . . . . . . 7 | |
63 | 62 | anbi1d 460 | . . . . . 6 |
64 | 61, 63 | anbi12d 464 | . . . . 5 |
65 | 64 | ceqsexgv 2809 | . . . 4 |
66 | 59, 65 | syl 14 | . . 3 |
67 | 12, 55, 66 | 3bitrd 213 | . 2 |
68 | 1, 10, 67 | pm5.21nii 693 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cvv 2681 csn 3522 cop 3525 cuni 3731 cxp 4532 cdm 4534 crn 4535 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-dm 4544 df-rn 4545 |
This theorem is referenced by: elxp6 6060 xpdom2 6718 |
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