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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 4828 when the double intersection does not create class existence problems (caused by int0 3650). (Contributed by NM, 1-Aug-2004.) |
Ref | Expression |
---|---|
elxp5 |
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 | sneq 3409 | . . . . . . . . . . . . . 14 | |
13 | 12 | rneqd 4581 | . . . . . . . . . . . . 13 |
14 | 13 | unieqd 3612 | . . . . . . . . . . . 12 |
15 | vex 2604 | . . . . . . . . . . . . 13 | |
16 | vex 2604 | . . . . . . . . . . . . 13 | |
17 | 15, 16 | op2nda 4825 | . . . . . . . . . . . 12 |
18 | 14, 17 | syl6req 2130 | . . . . . . . . . . 11 |
19 | 18 | pm4.71ri 384 | . . . . . . . . . 10 |
20 | 19 | anbi1i 445 | . . . . . . . . 9 |
21 | anass 393 | . . . . . . . . 9 | |
22 | 20, 21 | bitri 182 | . . . . . . . 8 |
23 | 22 | exbii 1536 | . . . . . . 7 |
24 | snexg 3956 | . . . . . . . . . 10 | |
25 | rnexg 4615 | . . . . . . . . . 10 | |
26 | 24, 25 | syl 14 | . . . . . . . . 9 |
27 | uniexg 4193 | . . . . . . . . 9 | |
28 | 26, 27 | syl 14 | . . . . . . . 8 |
29 | opeq2 3571 | . . . . . . . . . . 11 | |
30 | 29 | eqeq2d 2092 | . . . . . . . . . 10 |
31 | eleq1 2141 | . . . . . . . . . . 11 | |
32 | 31 | anbi2d 451 | . . . . . . . . . 10 |
33 | 30, 32 | anbi12d 456 | . . . . . . . . 9 |
34 | 33 | ceqsexgv 2724 | . . . . . . . 8 |
35 | 28, 34 | syl 14 | . . . . . . 7 |
36 | 23, 35 | syl5bb 190 | . . . . . 6 |
37 | inteq 3639 | . . . . . . . . . . . 12 | |
38 | 37 | inteqd 3641 | . . . . . . . . . . 11 |
39 | 38 | adantl 271 | . . . . . . . . . 10 |
40 | op1stbg 4228 | . . . . . . . . . . . 12 | |
41 | 15, 28, 40 | sylancr 405 | . . . . . . . . . . 11 |
42 | 41 | adantr 270 | . . . . . . . . . 10 |
43 | 39, 42 | eqtr2d 2114 | . . . . . . . . 9 |
44 | 43 | ex 113 | . . . . . . . 8 |
45 | 44 | pm4.71rd 386 | . . . . . . 7 |
46 | 45 | anbi1d 452 | . . . . . 6 |
47 | anass 393 | . . . . . . 7 | |
48 | 47 | a1i 9 | . . . . . 6 |
49 | 36, 46, 48 | 3bitrd 212 | . . . . 5 |
50 | 49 | exbidv 1746 | . . . 4 |
51 | 11, 50 | syl5bb 190 | . . 3 |
52 | eqvisset 2609 | . . . . . 6 | |
53 | 52 | adantr 270 | . . . . 5 |
54 | 53 | exlimiv 1529 | . . . 4 |
55 | 2 | ad2antrl 473 | . . . 4 |
56 | opeq1 3570 | . . . . . . 7 | |
57 | 56 | eqeq2d 2092 | . . . . . 6 |
58 | eleq1 2141 | . . . . . . 7 | |
59 | 58 | anbi1d 452 | . . . . . 6 |
60 | 57, 59 | anbi12d 456 | . . . . 5 |
61 | 60 | ceqsexgv 2724 | . . . 4 |
62 | 54, 55, 61 | pm5.21nii 652 | . . 3 |
63 | 51, 62 | syl6bb 194 | . 2 |
64 | 1, 10, 63 | 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 cint 3636 cxp 4361 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-int 3637 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: (None) |
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