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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 5109. (Contributed by NM, 17-Feb-2004.) |
Ref | Expression |
---|---|
elxp4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2746 | . 2 | |
2 | elex 2746 | . . . 4 | |
3 | elex 2746 | . . . 4 | |
4 | 2, 3 | anim12i 338 | . . 3 |
5 | opexg 4222 | . . . . 5 | |
6 | 5 | adantl 277 | . . . 4 |
7 | eleq1 2238 | . . . . 5 | |
8 | 7 | adantr 276 | . . . 4 |
9 | 6, 8 | mpbird 167 | . . 3 |
10 | 4, 9 | sylan2 286 | . 2 |
11 | elxp 4637 | . . . 4 | |
12 | 11 | a1i 9 | . . 3 |
13 | sneq 3600 | . . . . . . . . . . . . 13 | |
14 | 13 | rneqd 4849 | . . . . . . . . . . . 12 |
15 | 14 | unieqd 3816 | . . . . . . . . . . 11 |
16 | vex 2738 | . . . . . . . . . . . 12 | |
17 | vex 2738 | . . . . . . . . . . . 12 | |
18 | 16, 17 | op2nda 5105 | . . . . . . . . . . 11 |
19 | 15, 18 | eqtr2di 2225 | . . . . . . . . . 10 |
20 | 19 | pm4.71ri 392 | . . . . . . . . 9 |
21 | 20 | anbi1i 458 | . . . . . . . 8 |
22 | anass 401 | . . . . . . . 8 | |
23 | 21, 22 | bitri 184 | . . . . . . 7 |
24 | 23 | exbii 1603 | . . . . . 6 |
25 | snexg 4179 | . . . . . . . . 9 | |
26 | rnexg 4885 | . . . . . . . . 9 | |
27 | 25, 26 | syl 14 | . . . . . . . 8 |
28 | uniexg 4433 | . . . . . . . 8 | |
29 | 27, 28 | syl 14 | . . . . . . 7 |
30 | opeq2 3775 | . . . . . . . . . 10 | |
31 | 30 | eqeq2d 2187 | . . . . . . . . 9 |
32 | eleq1 2238 | . . . . . . . . . 10 | |
33 | 32 | anbi2d 464 | . . . . . . . . 9 |
34 | 31, 33 | anbi12d 473 | . . . . . . . 8 |
35 | 34 | ceqsexgv 2864 | . . . . . . 7 |
36 | 29, 35 | syl 14 | . . . . . 6 |
37 | 24, 36 | bitrid 192 | . . . . 5 |
38 | sneq 3600 | . . . . . . . . . . . 12 | |
39 | 38 | dmeqd 4822 | . . . . . . . . . . 11 |
40 | 39 | unieqd 3816 | . . . . . . . . . 10 |
41 | 40 | adantl 277 | . . . . . . . . 9 |
42 | dmsnopg 5092 | . . . . . . . . . . . . 13 | |
43 | 29, 42 | syl 14 | . . . . . . . . . . . 12 |
44 | 43 | unieqd 3816 | . . . . . . . . . . 11 |
45 | 16 | unisn 3821 | . . . . . . . . . . 11 |
46 | 44, 45 | eqtrdi 2224 | . . . . . . . . . 10 |
47 | 46 | adantr 276 | . . . . . . . . 9 |
48 | 41, 47 | eqtr2d 2209 | . . . . . . . 8 |
49 | 48 | ex 115 | . . . . . . 7 |
50 | 49 | pm4.71rd 394 | . . . . . 6 |
51 | 50 | anbi1d 465 | . . . . 5 |
52 | anass 401 | . . . . . 6 | |
53 | 52 | a1i 9 | . . . . 5 |
54 | 37, 51, 53 | 3bitrd 214 | . . . 4 |
55 | 54 | exbidv 1823 | . . 3 |
56 | dmexg 4884 | . . . . . 6 | |
57 | 25, 56 | syl 14 | . . . . 5 |
58 | uniexg 4433 | . . . . 5 | |
59 | 57, 58 | syl 14 | . . . 4 |
60 | opeq1 3774 | . . . . . . 7 | |
61 | 60 | eqeq2d 2187 | . . . . . 6 |
62 | eleq1 2238 | . . . . . . 7 | |
63 | 62 | anbi1d 465 | . . . . . 6 |
64 | 61, 63 | anbi12d 473 | . . . . 5 |
65 | 64 | ceqsexgv 2864 | . . . 4 |
66 | 59, 65 | syl 14 | . . 3 |
67 | 12, 55, 66 | 3bitrd 214 | . 2 |
68 | 1, 10, 67 | pm5.21nii 704 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 104 wb 105 wceq 1353 wex 1490 wcel 2146 cvv 2735 csn 3589 cop 3592 cuni 3805 cxp 4618 cdm 4620 crn 4621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-v 2737 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-xp 4626 df-rel 4627 df-cnv 4628 df-dm 4630 df-rn 4631 |
This theorem is referenced by: elxp6 6160 xpdom2 6821 |
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