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Mirrors > Home > ILE Home > Th. List > dfiin2g | Unicode version |
Description: Alternate definition of indexed intersection when is a set. (Contributed by Jeff Hankins, 27-Aug-2009.) |
Ref | Expression |
---|---|
dfiin2g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2421 | . . . 4 | |
2 | df-ral 2421 | . . . . . 6 | |
3 | eleq2 2203 | . . . . . . . . . . . . 13 | |
4 | 3 | biimprcd 159 | . . . . . . . . . . . 12 |
5 | 4 | alrimiv 1846 | . . . . . . . . . . 11 |
6 | eqid 2139 | . . . . . . . . . . . 12 | |
7 | eqeq1 2146 | . . . . . . . . . . . . . 14 | |
8 | 7, 3 | imbi12d 233 | . . . . . . . . . . . . 13 |
9 | 8 | spcgv 2773 | . . . . . . . . . . . 12 |
10 | 6, 9 | mpii 44 | . . . . . . . . . . 11 |
11 | 5, 10 | impbid2 142 | . . . . . . . . . 10 |
12 | 11 | imim2i 12 | . . . . . . . . 9 |
13 | 12 | pm5.74d 181 | . . . . . . . 8 |
14 | 13 | alimi 1431 | . . . . . . 7 |
15 | albi 1444 | . . . . . . 7 | |
16 | 14, 15 | syl 14 | . . . . . 6 |
17 | 2, 16 | sylbi 120 | . . . . 5 |
18 | df-ral 2421 | . . . . . . . 8 | |
19 | 18 | albii 1446 | . . . . . . 7 |
20 | alcom 1454 | . . . . . . 7 | |
21 | 19, 20 | bitr4i 186 | . . . . . 6 |
22 | r19.23v 2541 | . . . . . . . 8 | |
23 | vex 2689 | . . . . . . . . . 10 | |
24 | eqeq1 2146 | . . . . . . . . . . 11 | |
25 | 24 | rexbidv 2438 | . . . . . . . . . 10 |
26 | 23, 25 | elab 2828 | . . . . . . . . 9 |
27 | 26 | imbi1i 237 | . . . . . . . 8 |
28 | 22, 27 | bitr4i 186 | . . . . . . 7 |
29 | 28 | albii 1446 | . . . . . 6 |
30 | 19.21v 1845 | . . . . . . 7 | |
31 | 30 | albii 1446 | . . . . . 6 |
32 | 21, 29, 31 | 3bitr3ri 210 | . . . . 5 |
33 | 17, 32 | syl6bb 195 | . . . 4 |
34 | 1, 33 | syl5bb 191 | . . 3 |
35 | 34 | abbidv 2257 | . 2 |
36 | df-iin 3816 | . 2 | |
37 | df-int 3772 | . 2 | |
38 | 35, 36, 37 | 3eqtr4g 2197 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wal 1329 wceq 1331 wcel 1480 cab 2125 wral 2416 wrex 2417 cint 3771 ciin 3814 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-int 3772 df-iin 3816 |
This theorem is referenced by: dfiin2 3848 iinexgm 4079 dfiin3g 4797 fniinfv 5479 |
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