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Mirrors > Home > ILE Home > Th. List > iunin2 | Unicode version |
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3836 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) |
Ref | Expression |
---|---|
iunin2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.42v 2565 | . . . 4 | |
2 | elin 3229 | . . . . 5 | |
3 | 2 | rexbii 2419 | . . . 4 |
4 | eliun 3787 | . . . . 5 | |
5 | 4 | anbi2i 452 | . . . 4 |
6 | 1, 3, 5 | 3bitr4i 211 | . . 3 |
7 | eliun 3787 | . . 3 | |
8 | elin 3229 | . . 3 | |
9 | 6, 7, 8 | 3bitr4i 211 | . 2 |
10 | 9 | eqriv 2114 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1316 wcel 1465 wrex 2394 cin 3040 ciun 3783 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-in 3047 df-iun 3785 |
This theorem is referenced by: iunin1 3847 2iunin 3849 resiun1 4808 resiun2 4809 |
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