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Mirrors > Home > ILE Home > Th. List > inuni | Unicode version |
Description: The intersection of a union with a class is equal to the union of the intersections of each element of with . (Contributed by FL, 24-Mar-2007.) |
Ref | Expression |
---|---|
inuni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni2 3740 | . . . . 5 | |
2 | 1 | anbi1i 453 | . . . 4 |
3 | elin 3259 | . . . 4 | |
4 | ancom 264 | . . . . . . . 8 | |
5 | r19.41v 2587 | . . . . . . . 8 | |
6 | 4, 5 | bitr4i 186 | . . . . . . 7 |
7 | 6 | exbii 1584 | . . . . . 6 |
8 | rexcom4 2709 | . . . . . 6 | |
9 | 7, 8 | bitr4i 186 | . . . . 5 |
10 | vex 2689 | . . . . . . . . . 10 | |
11 | 10 | inex1 4062 | . . . . . . . . 9 |
12 | eleq2 2203 | . . . . . . . . 9 | |
13 | 11, 12 | ceqsexv 2725 | . . . . . . . 8 |
14 | elin 3259 | . . . . . . . 8 | |
15 | 13, 14 | bitri 183 | . . . . . . 7 |
16 | 15 | rexbii 2442 | . . . . . 6 |
17 | r19.41v 2587 | . . . . . 6 | |
18 | 16, 17 | bitri 183 | . . . . 5 |
19 | 9, 18 | bitri 183 | . . . 4 |
20 | 2, 3, 19 | 3bitr4i 211 | . . 3 |
21 | eluniab 3748 | . . 3 | |
22 | 20, 21 | bitr4i 186 | . 2 |
23 | 22 | eqriv 2136 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1331 wex 1468 wcel 1480 cab 2125 wrex 2417 cin 3070 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 ax-sep 4046 |
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-rex 2422 df-v 2688 df-in 3077 df-uni 3737 |
This theorem is referenced by: (None) |
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