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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 5096 when the double intersection does not create class existence problems (caused by int0 3843). (Contributed by NM, 1-Aug-2004.) |
Ref | Expression |
---|---|
elxp5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2741 | . 2 | |
2 | elex 2741 | . . . 4 | |
3 | elex 2741 | . . . 4 | |
4 | 2, 3 | anim12i 336 | . . 3 |
5 | opexg 4211 | . . . . 5 | |
6 | 5 | adantl 275 | . . . 4 |
7 | eleq1 2233 | . . . . 5 | |
8 | 7 | adantr 274 | . . . 4 |
9 | 6, 8 | mpbird 166 | . . 3 |
10 | 4, 9 | sylan2 284 | . 2 |
11 | elxp 4626 | . . . 4 | |
12 | sneq 3592 | . . . . . . . . . . . . . 14 | |
13 | 12 | rneqd 4838 | . . . . . . . . . . . . 13 |
14 | 13 | unieqd 3805 | . . . . . . . . . . . 12 |
15 | vex 2733 | . . . . . . . . . . . . 13 | |
16 | vex 2733 | . . . . . . . . . . . . 13 | |
17 | 15, 16 | op2nda 5093 | . . . . . . . . . . . 12 |
18 | 14, 17 | eqtr2di 2220 | . . . . . . . . . . 11 |
19 | 18 | pm4.71ri 390 | . . . . . . . . . 10 |
20 | 19 | anbi1i 455 | . . . . . . . . 9 |
21 | anass 399 | . . . . . . . . 9 | |
22 | 20, 21 | bitri 183 | . . . . . . . 8 |
23 | 22 | exbii 1598 | . . . . . . 7 |
24 | snexg 4168 | . . . . . . . . . 10 | |
25 | rnexg 4874 | . . . . . . . . . 10 | |
26 | 24, 25 | syl 14 | . . . . . . . . 9 |
27 | uniexg 4422 | . . . . . . . . 9 | |
28 | 26, 27 | syl 14 | . . . . . . . 8 |
29 | opeq2 3764 | . . . . . . . . . . 11 | |
30 | 29 | eqeq2d 2182 | . . . . . . . . . 10 |
31 | eleq1 2233 | . . . . . . . . . . 11 | |
32 | 31 | anbi2d 461 | . . . . . . . . . 10 |
33 | 30, 32 | anbi12d 470 | . . . . . . . . 9 |
34 | 33 | ceqsexgv 2859 | . . . . . . . 8 |
35 | 28, 34 | syl 14 | . . . . . . 7 |
36 | 23, 35 | syl5bb 191 | . . . . . 6 |
37 | inteq 3832 | . . . . . . . . . . . 12 | |
38 | 37 | inteqd 3834 | . . . . . . . . . . 11 |
39 | 38 | adantl 275 | . . . . . . . . . 10 |
40 | op1stbg 4462 | . . . . . . . . . . . 12 | |
41 | 15, 28, 40 | sylancr 412 | . . . . . . . . . . 11 |
42 | 41 | adantr 274 | . . . . . . . . . 10 |
43 | 39, 42 | eqtr2d 2204 | . . . . . . . . 9 |
44 | 43 | ex 114 | . . . . . . . 8 |
45 | 44 | pm4.71rd 392 | . . . . . . 7 |
46 | 45 | anbi1d 462 | . . . . . 6 |
47 | anass 399 | . . . . . . 7 | |
48 | 47 | a1i 9 | . . . . . 6 |
49 | 36, 46, 48 | 3bitrd 213 | . . . . 5 |
50 | 49 | exbidv 1818 | . . . 4 |
51 | 11, 50 | syl5bb 191 | . . 3 |
52 | eqvisset 2740 | . . . . . 6 | |
53 | 52 | adantr 274 | . . . . 5 |
54 | 53 | exlimiv 1591 | . . . 4 |
55 | 2 | ad2antrl 487 | . . . 4 |
56 | opeq1 3763 | . . . . . . 7 | |
57 | 56 | eqeq2d 2182 | . . . . . 6 |
58 | eleq1 2233 | . . . . . . 7 | |
59 | 58 | anbi1d 462 | . . . . . 6 |
60 | 57, 59 | anbi12d 470 | . . . . 5 |
61 | 60 | ceqsexgv 2859 | . . . 4 |
62 | 54, 55, 61 | pm5.21nii 699 | . . 3 |
63 | 51, 62 | bitrdi 195 | . 2 |
64 | 1, 10, 63 | pm5.21nii 699 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1348 wex 1485 wcel 2141 cvv 2730 csn 3581 cop 3584 cuni 3794 cint 3829 cxp 4607 crn 4610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-xp 4615 df-rel 4616 df-cnv 4617 df-dm 4619 df-rn 4620 |
This theorem is referenced by: (None) |
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