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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 2422 | . . . 4 | |
2 | vex 2560 | . . . . . 6 | |
3 | eliin 3662 | . . . . . 6 | |
4 | 2, 3 | ax-mp 7 | . . . . 5 |
5 | 4 | rexbii 2331 | . . . 4 |
6 | eliun 3661 | . . . . 5 | |
7 | 6 | ralbii 2330 | . . . 4 |
8 | 1, 5, 7 | 3imtr4i 190 | . . 3 |
9 | eliun 3661 | . . 3 | |
10 | eliin 3662 | . . . 4 | |
11 | 2, 10 | ax-mp 7 | . . 3 |
12 | 8, 9, 11 | 3imtr4i 190 | . 2 |
13 | 12 | ssriv 2949 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 98 wcel 1393 wral 2306 wrex 2307 cvv 2557 wss 2917 ciun 3657 ciin 3658 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-in 2924 df-ss 2931 df-iun 3659 df-iin 3660 |
This theorem is referenced by: (None) |
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