Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3913 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) |
Ref | Expression |
---|---|
iunin2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.42v 2621 | . . . 4 | |
2 | elin 3300 | . . . . 5 | |
3 | 2 | rexbii 2471 | . . . 4 |
4 | eliun 3864 | . . . . 5 | |
5 | 4 | anbi2i 453 | . . . 4 |
6 | 1, 3, 5 | 3bitr4i 211 | . . 3 |
7 | eliun 3864 | . . 3 | |
8 | elin 3300 | . . 3 | |
9 | 6, 7, 8 | 3bitr4i 211 | . 2 |
10 | 9 | eqriv 2161 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1342 wcel 2135 wrex 2443 cin 3110 ciun 3860 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-in 3117 df-iun 3862 |
This theorem is referenced by: iunin1 3924 2iunin 3926 resiun1 4897 resiun2 4898 |
Copyright terms: Public domain | W3C validator |