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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 3605 | . . . . 5 | |
2 | 1 | anbi1i 445 | . . . 4 |
3 | elin 3155 | . . . 4 | |
4 | ancom 262 | . . . . . . . 8 | |
5 | r19.41v 2510 | . . . . . . . 8 | |
6 | 4, 5 | bitr4i 185 | . . . . . . 7 |
7 | 6 | exbii 1536 | . . . . . 6 |
8 | rexcom4 2622 | . . . . . 6 | |
9 | 7, 8 | bitr4i 185 | . . . . 5 |
10 | vex 2604 | . . . . . . . . . 10 | |
11 | 10 | inex1 3912 | . . . . . . . . 9 |
12 | eleq2 2142 | . . . . . . . . 9 | |
13 | 11, 12 | ceqsexv 2638 | . . . . . . . 8 |
14 | elin 3155 | . . . . . . . 8 | |
15 | 13, 14 | bitri 182 | . . . . . . 7 |
16 | 15 | rexbii 2373 | . . . . . 6 |
17 | r19.41v 2510 | . . . . . 6 | |
18 | 16, 17 | bitri 182 | . . . . 5 |
19 | 9, 18 | bitri 182 | . . . 4 |
20 | 2, 3, 19 | 3bitr4i 210 | . . 3 |
21 | eluniab 3613 | . . 3 | |
22 | 20, 21 | bitr4i 185 | . 2 |
23 | 22 | eqriv 2078 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wceq 1284 wex 1421 wcel 1433 cab 2067 wrex 2349 cin 2972 cuni 3601 |
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 ax-sep 3896 |
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-rex 2354 df-v 2603 df-in 2979 df-uni 3602 |
This theorem is referenced by: (None) |
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