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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 4829. (Contributed by NM, 17-Feb-2004.) |
Ref | Expression |
---|---|
elxp4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2610 | . 2 | |
2 | elex 2610 | . . . 4 | |
3 | elex 2610 | . . . 4 | |
4 | 2, 3 | anim12i 331 | . . 3 |
5 | opexg 3983 | . . . . 5 | |
6 | 5 | adantl 271 | . . . 4 |
7 | eleq1 2141 | . . . . 5 | |
8 | 7 | adantr 270 | . . . 4 |
9 | 6, 8 | mpbird 165 | . . 3 |
10 | 4, 9 | sylan2 280 | . 2 |
11 | elxp 4380 | . . . 4 | |
12 | 11 | a1i 9 | . . 3 |
13 | sneq 3409 | . . . . . . . . . . . . 13 | |
14 | 13 | rneqd 4581 | . . . . . . . . . . . 12 |
15 | 14 | unieqd 3612 | . . . . . . . . . . 11 |
16 | vex 2604 | . . . . . . . . . . . 12 | |
17 | vex 2604 | . . . . . . . . . . . 12 | |
18 | 16, 17 | op2nda 4825 | . . . . . . . . . . 11 |
19 | 15, 18 | syl6req 2130 | . . . . . . . . . 10 |
20 | 19 | pm4.71ri 384 | . . . . . . . . 9 |
21 | 20 | anbi1i 445 | . . . . . . . 8 |
22 | anass 393 | . . . . . . . 8 | |
23 | 21, 22 | bitri 182 | . . . . . . 7 |
24 | 23 | exbii 1536 | . . . . . 6 |
25 | snexg 3956 | . . . . . . . . 9 | |
26 | rnexg 4615 | . . . . . . . . 9 | |
27 | 25, 26 | syl 14 | . . . . . . . 8 |
28 | uniexg 4193 | . . . . . . . 8 | |
29 | 27, 28 | syl 14 | . . . . . . 7 |
30 | opeq2 3571 | . . . . . . . . . 10 | |
31 | 30 | eqeq2d 2092 | . . . . . . . . 9 |
32 | eleq1 2141 | . . . . . . . . . 10 | |
33 | 32 | anbi2d 451 | . . . . . . . . 9 |
34 | 31, 33 | anbi12d 456 | . . . . . . . 8 |
35 | 34 | ceqsexgv 2724 | . . . . . . 7 |
36 | 29, 35 | syl 14 | . . . . . 6 |
37 | 24, 36 | syl5bb 190 | . . . . 5 |
38 | sneq 3409 | . . . . . . . . . . . 12 | |
39 | 38 | dmeqd 4555 | . . . . . . . . . . 11 |
40 | 39 | unieqd 3612 | . . . . . . . . . 10 |
41 | 40 | adantl 271 | . . . . . . . . 9 |
42 | dmsnopg 4812 | . . . . . . . . . . . . 13 | |
43 | 29, 42 | syl 14 | . . . . . . . . . . . 12 |
44 | 43 | unieqd 3612 | . . . . . . . . . . 11 |
45 | 16 | unisn 3617 | . . . . . . . . . . 11 |
46 | 44, 45 | syl6eq 2129 | . . . . . . . . . 10 |
47 | 46 | adantr 270 | . . . . . . . . 9 |
48 | 41, 47 | eqtr2d 2114 | . . . . . . . 8 |
49 | 48 | ex 113 | . . . . . . 7 |
50 | 49 | pm4.71rd 386 | . . . . . 6 |
51 | 50 | anbi1d 452 | . . . . 5 |
52 | anass 393 | . . . . . 6 | |
53 | 52 | a1i 9 | . . . . 5 |
54 | 37, 51, 53 | 3bitrd 212 | . . . 4 |
55 | 54 | exbidv 1746 | . . 3 |
56 | dmexg 4614 | . . . . . 6 | |
57 | 25, 56 | syl 14 | . . . . 5 |
58 | uniexg 4193 | . . . . 5 | |
59 | 57, 58 | syl 14 | . . . 4 |
60 | opeq1 3570 | . . . . . . 7 | |
61 | 60 | eqeq2d 2092 | . . . . . 6 |
62 | eleq1 2141 | . . . . . . 7 | |
63 | 62 | anbi1d 452 | . . . . . 6 |
64 | 61, 63 | anbi12d 456 | . . . . 5 |
65 | 64 | ceqsexgv 2724 | . . . 4 |
66 | 59, 65 | syl 14 | . . 3 |
67 | 12, 55, 66 | 3bitrd 212 | . 2 |
68 | 1, 10, 67 | pm5.21nii 652 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 cvv 2601 csn 3398 cop 3401 cuni 3601 cxp 4361 cdm 4363 crn 4364 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-dm 4373 df-rn 4374 |
This theorem is referenced by: elxp6 5816 xpdom2 6328 |
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