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Mirrors > Home > NFE Home > Th. List > uniin | Unicode version |
Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs in set.mm for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
uniin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.40 1609 | . . . 4 | |
2 | elin 3219 | . . . . . . 7 | |
3 | 2 | anbi2i 675 | . . . . . 6 |
4 | anandi 801 | . . . . . 6 | |
5 | 3, 4 | bitri 240 | . . . . 5 |
6 | 5 | exbii 1582 | . . . 4 |
7 | eluni 3894 | . . . . 5 | |
8 | eluni 3894 | . . . . 5 | |
9 | 7, 8 | anbi12i 678 | . . . 4 |
10 | 1, 6, 9 | 3imtr4i 257 | . . 3 |
11 | eluni 3894 | . . 3 | |
12 | elin 3219 | . . 3 | |
13 | 10, 11, 12 | 3imtr4i 257 | . 2 |
14 | 13 | ssriv 3277 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 358 wex 1541 wcel 1710 cin 3208 wss 3257 cuni 3891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-uni 3892 |
This theorem is referenced by: (None) |
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