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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 3121 | . . . . . . . 8 | |
2 | 1 | biimprcd 159 | . . . . . . 7 |
3 | 2 | reximdv 2533 | . . . . . 6 |
4 | 3 | com12 30 | . . . . 5 |
5 | ssiun 3855 | . . . . . 6 | |
6 | uniiun 3866 | . . . . . 6 | |
7 | 5, 6 | sseqtrrdi 3146 | . . . . 5 |
8 | 4, 7 | impbid1 141 | . . . 4 |
9 | vex 2689 | . . . . 5 | |
10 | 9 | elpw 3516 | . . . 4 |
11 | eliun 3817 | . . . . 5 | |
12 | df-pw 3512 | . . . . . . 7 | |
13 | 12 | abeq2i 2250 | . . . . . 6 |
14 | 13 | rexbii 2442 | . . . . 5 |
15 | 11, 14 | bitri 183 | . . . 4 |
16 | 8, 10, 15 | 3bitr4g 222 | . . 3 |
17 | 16 | eqrdv 2137 | . 2 |
18 | ssid 3117 | . . . . 5 | |
19 | iunpw.1 | . . . . . . . 8 | |
20 | 19 | uniex 4359 | . . . . . . 7 |
21 | 20 | elpw 3516 | . . . . . 6 |
22 | eleq2 2203 | . . . . . 6 | |
23 | 21, 22 | syl5bbr 193 | . . . . 5 |
24 | 18, 23 | mpbii 147 | . . . 4 |
25 | eliun 3817 | . . . 4 | |
26 | 24, 25 | sylib 121 | . . 3 |
27 | elssuni 3764 | . . . . . . 7 | |
28 | elpwi 3519 | . . . . . . 7 | |
29 | 27, 28 | anim12i 336 | . . . . . 6 |
30 | eqss 3112 | . . . . . 6 | |
31 | 29, 30 | sylibr 133 | . . . . 5 |
32 | 31 | ex 114 | . . . 4 |
33 | 32 | reximia 2527 | . . 3 |
34 | 26, 33 | syl 14 | . 2 |
35 | 17, 34 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wcel 1480 wrex 2417 cvv 2686 wss 3071 cpw 3510 cuni 3736 ciun 3813 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-un 4355 |
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-in 3077 df-ss 3084 df-pw 3512 df-uni 3737 df-iun 3815 |
This theorem is referenced by: (None) |
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