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Mirrors > Home > ILE Home > Th. List > iunpw | Unicode version |
Description: An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.) |
Ref | Expression |
---|---|
iunpw.1 |
Ref | Expression |
---|---|
iunpw |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 2967 | . . . . . . . 8 | |
2 | 1 | biimprcd 149 | . . . . . . 7 |
3 | 2 | reximdv 2420 | . . . . . 6 |
4 | 3 | com12 27 | . . . . 5 |
5 | ssiun 3699 | . . . . . 6 | |
6 | uniiun 3710 | . . . . . 6 | |
7 | 5, 6 | syl6sseqr 2992 | . . . . 5 |
8 | 4, 7 | impbid1 130 | . . . 4 |
9 | vex 2560 | . . . . 5 | |
10 | 9 | elpw 3365 | . . . 4 |
11 | eliun 3661 | . . . . 5 | |
12 | df-pw 3361 | . . . . . . 7 | |
13 | 12 | abeq2i 2148 | . . . . . 6 |
14 | 13 | rexbii 2331 | . . . . 5 |
15 | 11, 14 | bitri 173 | . . . 4 |
16 | 8, 10, 15 | 3bitr4g 212 | . . 3 |
17 | 16 | eqrdv 2038 | . 2 |
18 | ssid 2964 | . . . . 5 | |
19 | iunpw.1 | . . . . . . . 8 | |
20 | 19 | uniex 4174 | . . . . . . 7 |
21 | 20 | elpw 3365 | . . . . . 6 |
22 | eleq2 2101 | . . . . . 6 | |
23 | 21, 22 | syl5bbr 183 | . . . . 5 |
24 | 18, 23 | mpbii 136 | . . . 4 |
25 | eliun 3661 | . . . 4 | |
26 | 24, 25 | sylib 127 | . . 3 |
27 | elssuni 3608 | . . . . . . 7 | |
28 | elpwi 3368 | . . . . . . 7 | |
29 | 27, 28 | anim12i 321 | . . . . . 6 |
30 | eqss 2960 | . . . . . 6 | |
31 | 29, 30 | sylibr 137 | . . . . 5 |
32 | 31 | ex 108 | . . . 4 |
33 | 32 | reximia 2414 | . . 3 |
34 | 26, 33 | syl 14 | . 2 |
35 | 17, 34 | impbii 117 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wcel 1393 wrex 2307 cvv 2557 wss 2917 cpw 3359 cuni 3580 ciun 3657 |
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-un 4170 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-in 2924 df-ss 2931 df-pw 3361 df-uni 3581 df-iun 3659 |
This theorem is referenced by: (None) |
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