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Mirrors > Home > ILE Home > Th. List > elxp5 | Unicode version |
Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4808 when the double intersection does not create class existence problems (caused by int0 3629). (Contributed by NM, 1-Aug-2004.) |
Ref | Expression |
---|---|
elxp5 |
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 | sneq 3386 | . . . . . . . . . . . . . 14 | |
13 | 12 | rneqd 4563 | . . . . . . . . . . . . 13 |
14 | 13 | unieqd 3591 | . . . . . . . . . . . 12 |
15 | vex 2560 | . . . . . . . . . . . . 13 | |
16 | vex 2560 | . . . . . . . . . . . . 13 | |
17 | 15, 16 | op2nda 4805 | . . . . . . . . . . . 12 |
18 | 14, 17 | syl6req 2089 | . . . . . . . . . . 11 |
19 | 18 | pm4.71ri 372 | . . . . . . . . . 10 |
20 | 19 | anbi1i 431 | . . . . . . . . 9 |
21 | anass 381 | . . . . . . . . 9 | |
22 | 20, 21 | bitri 173 | . . . . . . . 8 |
23 | 22 | exbii 1496 | . . . . . . 7 |
24 | snexgOLD 3935 | . . . . . . . . . 10 | |
25 | rnexg 4597 | . . . . . . . . . 10 | |
26 | 24, 25 | syl 14 | . . . . . . . . 9 |
27 | uniexg 4175 | . . . . . . . . 9 | |
28 | 26, 27 | syl 14 | . . . . . . . 8 |
29 | opeq2 3550 | . . . . . . . . . . 11 | |
30 | 29 | eqeq2d 2051 | . . . . . . . . . 10 |
31 | eleq1 2100 | . . . . . . . . . . 11 | |
32 | 31 | anbi2d 437 | . . . . . . . . . 10 |
33 | 30, 32 | anbi12d 442 | . . . . . . . . 9 |
34 | 33 | ceqsexgv 2673 | . . . . . . . 8 |
35 | 28, 34 | syl 14 | . . . . . . 7 |
36 | 23, 35 | syl5bb 181 | . . . . . 6 |
37 | inteq 3618 | . . . . . . . . . . . 12 | |
38 | 37 | inteqd 3620 | . . . . . . . . . . 11 |
39 | 38 | adantl 262 | . . . . . . . . . 10 |
40 | op1stbg 4210 | . . . . . . . . . . . 12 | |
41 | 15, 28, 40 | sylancr 393 | . . . . . . . . . . 11 |
42 | 41 | adantr 261 | . . . . . . . . . 10 |
43 | 39, 42 | eqtr2d 2073 | . . . . . . . . 9 |
44 | 43 | ex 108 | . . . . . . . 8 |
45 | 44 | pm4.71rd 374 | . . . . . . 7 |
46 | 45 | anbi1d 438 | . . . . . 6 |
47 | anass 381 | . . . . . . 7 | |
48 | 47 | a1i 9 | . . . . . 6 |
49 | 36, 46, 48 | 3bitrd 203 | . . . . 5 |
50 | 49 | exbidv 1706 | . . . 4 |
51 | 11, 50 | syl5bb 181 | . . 3 |
52 | eleq1 2100 | . . . . . . 7 | |
53 | 15, 52 | mpbii 136 | . . . . . 6 |
54 | 53 | adantr 261 | . . . . 5 |
55 | 54 | exlimiv 1489 | . . . 4 |
56 | 2 | ad2antrl 459 | . . . 4 |
57 | opeq1 3549 | . . . . . . 7 | |
58 | 57 | eqeq2d 2051 | . . . . . 6 |
59 | eleq1 2100 | . . . . . . 7 | |
60 | 59 | anbi1d 438 | . . . . . 6 |
61 | 58, 60 | anbi12d 442 | . . . . 5 |
62 | 61 | ceqsexgv 2673 | . . . 4 |
63 | 55, 56, 62 | pm5.21nii 620 | . . 3 |
64 | 51, 63 | syl6bb 185 | . 2 |
65 | 1, 10, 64 | 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 cint 3615 cxp 4343 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-int 3616 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: (None) |
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