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Mirrors > Home > ILE Home > Th. List > uniin | Unicode version |
Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
uniin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.40 1610 | . . . 4 | |
2 | elin 3259 | . . . . . . 7 | |
3 | 2 | anbi2i 452 | . . . . . 6 |
4 | anandi 579 | . . . . . 6 | |
5 | 3, 4 | bitri 183 | . . . . 5 |
6 | 5 | exbii 1584 | . . . 4 |
7 | eluni 3739 | . . . . 5 | |
8 | eluni 3739 | . . . . 5 | |
9 | 7, 8 | anbi12i 455 | . . . 4 |
10 | 1, 6, 9 | 3imtr4i 200 | . . 3 |
11 | eluni 3739 | . . 3 | |
12 | elin 3259 | . . 3 | |
13 | 10, 11, 12 | 3imtr4i 200 | . 2 |
14 | 13 | ssriv 3101 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wex 1468 wcel 1480 cin 3070 wss 3071 cuni 3736 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-in 3077 df-ss 3084 df-uni 3737 |
This theorem is referenced by: tgval 12218 |
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