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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 2576 | . . . 4 | |
2 | vex 2733 | . . . . . 6 | |
3 | eliin 3876 | . . . . . 6 | |
4 | 2, 3 | ax-mp 5 | . . . . 5 |
5 | 4 | rexbii 2477 | . . . 4 |
6 | eliun 3875 | . . . . 5 | |
7 | 6 | ralbii 2476 | . . . 4 |
8 | 1, 5, 7 | 3imtr4i 200 | . . 3 |
9 | eliun 3875 | . . 3 | |
10 | eliin 3876 | . . . 4 | |
11 | 2, 10 | ax-mp 5 | . . 3 |
12 | 8, 9, 11 | 3imtr4i 200 | . 2 |
13 | 12 | ssriv 3151 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 104 wcel 2141 wral 2448 wrex 2449 cvv 2730 wss 3121 ciun 3871 ciin 3872 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-in 3127 df-ss 3134 df-iun 3873 df-iin 3874 |
This theorem is referenced by: (None) |
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