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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 4809. (Contributed by NM, 17-Feb-2004.) |
Ref | Expression |
---|---|
elxp4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2566 | . 2 | |
2 | elex 2566 | . . . 4 | |
3 | elex 2566 | . . . 4 | |
4 | 2, 3 | anim12i 321 | . . 3 |
5 | opexgOLD 3965 | . . . . 5 | |
6 | 5 | adantl 262 | . . . 4 |
7 | eleq1 2100 | . . . . 5 | |
8 | 7 | adantr 261 | . . . 4 |
9 | 6, 8 | mpbird 156 | . . 3 |
10 | 4, 9 | sylan2 270 | . 2 |
11 | elxp 4362 | . . . 4 | |
12 | 11 | a1i 9 | . . 3 |
13 | sneq 3386 | . . . . . . . . . . . . 13 | |
14 | 13 | rneqd 4563 | . . . . . . . . . . . 12 |
15 | 14 | unieqd 3591 | . . . . . . . . . . 11 |
16 | vex 2560 | . . . . . . . . . . . 12 | |
17 | vex 2560 | . . . . . . . . . . . 12 | |
18 | 16, 17 | op2nda 4805 | . . . . . . . . . . 11 |
19 | 15, 18 | syl6req 2089 | . . . . . . . . . 10 |
20 | 19 | pm4.71ri 372 | . . . . . . . . 9 |
21 | 20 | anbi1i 431 | . . . . . . . 8 |
22 | anass 381 | . . . . . . . 8 | |
23 | 21, 22 | bitri 173 | . . . . . . 7 |
24 | 23 | exbii 1496 | . . . . . 6 |
25 | snexgOLD 3935 | . . . . . . . . 9 | |
26 | rnexg 4597 | . . . . . . . . 9 | |
27 | 25, 26 | syl 14 | . . . . . . . 8 |
28 | uniexg 4175 | . . . . . . . 8 | |
29 | 27, 28 | syl 14 | . . . . . . 7 |
30 | opeq2 3550 | . . . . . . . . . 10 | |
31 | 30 | eqeq2d 2051 | . . . . . . . . 9 |
32 | eleq1 2100 | . . . . . . . . . 10 | |
33 | 32 | anbi2d 437 | . . . . . . . . 9 |
34 | 31, 33 | anbi12d 442 | . . . . . . . 8 |
35 | 34 | ceqsexgv 2673 | . . . . . . 7 |
36 | 29, 35 | syl 14 | . . . . . 6 |
37 | 24, 36 | syl5bb 181 | . . . . 5 |
38 | sneq 3386 | . . . . . . . . . . . 12 | |
39 | 38 | dmeqd 4537 | . . . . . . . . . . 11 |
40 | 39 | unieqd 3591 | . . . . . . . . . 10 |
41 | 40 | adantl 262 | . . . . . . . . 9 |
42 | dmsnopg 4792 | . . . . . . . . . . . . 13 | |
43 | 29, 42 | syl 14 | . . . . . . . . . . . 12 |
44 | 43 | unieqd 3591 | . . . . . . . . . . 11 |
45 | 16 | unisn 3596 | . . . . . . . . . . 11 |
46 | 44, 45 | syl6eq 2088 | . . . . . . . . . 10 |
47 | 46 | adantr 261 | . . . . . . . . 9 |
48 | 41, 47 | eqtr2d 2073 | . . . . . . . 8 |
49 | 48 | ex 108 | . . . . . . 7 |
50 | 49 | pm4.71rd 374 | . . . . . 6 |
51 | 50 | anbi1d 438 | . . . . 5 |
52 | anass 381 | . . . . . 6 | |
53 | 52 | a1i 9 | . . . . 5 |
54 | 37, 51, 53 | 3bitrd 203 | . . . 4 |
55 | 54 | exbidv 1706 | . . 3 |
56 | dmexg 4596 | . . . . . 6 | |
57 | 25, 56 | syl 14 | . . . . 5 |
58 | uniexg 4175 | . . . . 5 | |
59 | 57, 58 | syl 14 | . . . 4 |
60 | opeq1 3549 | . . . . . . 7 | |
61 | 60 | eqeq2d 2051 | . . . . . 6 |
62 | eleq1 2100 | . . . . . . 7 | |
63 | 62 | anbi1d 438 | . . . . . 6 |
64 | 61, 63 | anbi12d 442 | . . . . 5 |
65 | 64 | ceqsexgv 2673 | . . . 4 |
66 | 59, 65 | syl 14 | . . 3 |
67 | 12, 55, 66 | 3bitrd 203 | . 2 |
68 | 1, 10, 67 | pm5.21nii 620 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cvv 2557 csn 3375 cop 3378 cuni 3580 cxp 4343 cdm 4345 crn 4346 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-dm 4355 df-rn 4356 |
This theorem is referenced by: elxp6 5796 xpdom2 6305 |
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