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Mirrors > Home > MPE Home > Th. List > iunin2 | Structured version Visualization version Unicode version |
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4573 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) |
Ref | Expression |
---|---|
iunin2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.42v 3092 | . . . 4 | |
2 | elin 3796 | . . . . 5 | |
3 | 2 | rexbii 3041 | . . . 4 |
4 | eliun 4524 | . . . . 5 | |
5 | 4 | anbi2i 730 | . . . 4 |
6 | 1, 3, 5 | 3bitr4i 292 | . . 3 |
7 | eliun 4524 | . . 3 | |
8 | elin 3796 | . . 3 | |
9 | 6, 7, 8 | 3bitr4i 292 | . 2 |
10 | 9 | eqriv 2619 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 384 wceq 1483 wcel 1990 wrex 2913 cin 3573 ciun 4520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-in 3581 df-iun 4522 |
This theorem is referenced by: iunin1 4585 2iunin 4588 resiun1OLD 5417 resiun2 5418 infssuni 8257 kmlem11 8982 cmpsublem 21202 cmpsub 21203 kgentopon 21341 metnrmlem3 22664 ovoliunlem1 23270 voliunlem1 23318 voliunlem2 23319 uniioombllem2 23351 uniioombllem4 23354 volsup2 23373 itg1addlem5 23467 itg1climres 23481 uniin2 29368 carsgclctunlem2 30381 cvmscld 31255 cnambfre 33458 ftc1anclem6 33490 heiborlem3 33612 carageniuncllem2 40736 |
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