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Mirrors > Home > ILE Home > Th. List > iuniin | Unicode version |
Description: Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
iuniin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.12 2466 | . . . 4 | |
2 | vex 2604 | . . . . . 6 | |
3 | eliin 3683 | . . . . . 6 | |
4 | 2, 3 | ax-mp 7 | . . . . 5 |
5 | 4 | rexbii 2373 | . . . 4 |
6 | eliun 3682 | . . . . 5 | |
7 | 6 | ralbii 2372 | . . . 4 |
8 | 1, 5, 7 | 3imtr4i 199 | . . 3 |
9 | eliun 3682 | . . 3 | |
10 | eliin 3683 | . . . 4 | |
11 | 2, 10 | ax-mp 7 | . . 3 |
12 | 8, 9, 11 | 3imtr4i 199 | . 2 |
13 | 12 | ssriv 3003 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 103 wcel 1433 wral 2348 wrex 2349 cvv 2601 wss 2973 ciun 3678 ciin 3679 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-in 2979 df-ss 2986 df-iun 3680 df-iin 3681 |
This theorem is referenced by: (None) |
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