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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 5086. (Contributed by NM, 17-Feb-2004.) |
Ref | Expression |
---|---|
elxp4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2732 | . 2 | |
2 | elex 2732 | . . . 4 | |
3 | elex 2732 | . . . 4 | |
4 | 2, 3 | anim12i 336 | . . 3 |
5 | opexg 4200 | . . . . 5 | |
6 | 5 | adantl 275 | . . . 4 |
7 | eleq1 2227 | . . . . 5 | |
8 | 7 | adantr 274 | . . . 4 |
9 | 6, 8 | mpbird 166 | . . 3 |
10 | 4, 9 | sylan2 284 | . 2 |
11 | elxp 4615 | . . . 4 | |
12 | 11 | a1i 9 | . . 3 |
13 | sneq 3581 | . . . . . . . . . . . . 13 | |
14 | 13 | rneqd 4827 | . . . . . . . . . . . 12 |
15 | 14 | unieqd 3794 | . . . . . . . . . . 11 |
16 | vex 2724 | . . . . . . . . . . . 12 | |
17 | vex 2724 | . . . . . . . . . . . 12 | |
18 | 16, 17 | op2nda 5082 | . . . . . . . . . . 11 |
19 | 15, 18 | eqtr2di 2214 | . . . . . . . . . 10 |
20 | 19 | pm4.71ri 390 | . . . . . . . . 9 |
21 | 20 | anbi1i 454 | . . . . . . . 8 |
22 | anass 399 | . . . . . . . 8 | |
23 | 21, 22 | bitri 183 | . . . . . . 7 |
24 | 23 | exbii 1592 | . . . . . 6 |
25 | snexg 4157 | . . . . . . . . 9 | |
26 | rnexg 4863 | . . . . . . . . 9 | |
27 | 25, 26 | syl 14 | . . . . . . . 8 |
28 | uniexg 4411 | . . . . . . . 8 | |
29 | 27, 28 | syl 14 | . . . . . . 7 |
30 | opeq2 3753 | . . . . . . . . . 10 | |
31 | 30 | eqeq2d 2176 | . . . . . . . . 9 |
32 | eleq1 2227 | . . . . . . . . . 10 | |
33 | 32 | anbi2d 460 | . . . . . . . . 9 |
34 | 31, 33 | anbi12d 465 | . . . . . . . 8 |
35 | 34 | ceqsexgv 2850 | . . . . . . 7 |
36 | 29, 35 | syl 14 | . . . . . 6 |
37 | 24, 36 | syl5bb 191 | . . . . 5 |
38 | sneq 3581 | . . . . . . . . . . . 12 | |
39 | 38 | dmeqd 4800 | . . . . . . . . . . 11 |
40 | 39 | unieqd 3794 | . . . . . . . . . 10 |
41 | 40 | adantl 275 | . . . . . . . . 9 |
42 | dmsnopg 5069 | . . . . . . . . . . . . 13 | |
43 | 29, 42 | syl 14 | . . . . . . . . . . . 12 |
44 | 43 | unieqd 3794 | . . . . . . . . . . 11 |
45 | 16 | unisn 3799 | . . . . . . . . . . 11 |
46 | 44, 45 | eqtrdi 2213 | . . . . . . . . . 10 |
47 | 46 | adantr 274 | . . . . . . . . 9 |
48 | 41, 47 | eqtr2d 2198 | . . . . . . . 8 |
49 | 48 | ex 114 | . . . . . . 7 |
50 | 49 | pm4.71rd 392 | . . . . . 6 |
51 | 50 | anbi1d 461 | . . . . 5 |
52 | anass 399 | . . . . . 6 | |
53 | 52 | a1i 9 | . . . . 5 |
54 | 37, 51, 53 | 3bitrd 213 | . . . 4 |
55 | 54 | exbidv 1812 | . . 3 |
56 | dmexg 4862 | . . . . . 6 | |
57 | 25, 56 | syl 14 | . . . . 5 |
58 | uniexg 4411 | . . . . 5 | |
59 | 57, 58 | syl 14 | . . . 4 |
60 | opeq1 3752 | . . . . . . 7 | |
61 | 60 | eqeq2d 2176 | . . . . . 6 |
62 | eleq1 2227 | . . . . . . 7 | |
63 | 62 | anbi1d 461 | . . . . . 6 |
64 | 61, 63 | anbi12d 465 | . . . . 5 |
65 | 64 | ceqsexgv 2850 | . . . 4 |
66 | 59, 65 | syl 14 | . . 3 |
67 | 12, 55, 66 | 3bitrd 213 | . 2 |
68 | 1, 10, 67 | pm5.21nii 694 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 cvv 2721 csn 3570 cop 3573 cuni 3783 cxp 4596 cdm 4598 crn 4599 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-dm 4608 df-rn 4609 |
This theorem is referenced by: elxp6 6129 xpdom2 6788 |
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