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Mirrors > Home > MPE Home > Th. List > uniin | Structured version Visualization version Unicode version |
Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 7827 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 1797 | . . . 4 | |
2 | elin 3796 | . . . . . . 7 | |
3 | 2 | anbi2i 730 | . . . . . 6 |
4 | anandi 871 | . . . . . 6 | |
5 | 3, 4 | bitri 264 | . . . . 5 |
6 | 5 | exbii 1774 | . . . 4 |
7 | eluni 4439 | . . . . 5 | |
8 | eluni 4439 | . . . . 5 | |
9 | 7, 8 | anbi12i 733 | . . . 4 |
10 | 1, 6, 9 | 3imtr4i 281 | . . 3 |
11 | eluni 4439 | . . 3 | |
12 | elin 3796 | . . 3 | |
13 | 10, 11, 12 | 3imtr4i 281 | . 2 |
14 | 13 | ssriv 3607 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 384 wex 1704 wcel 1990 cin 3573 wss 3574 cuni 4436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-ss 3588 df-uni 4437 |
This theorem is referenced by: uniinqs 7827 psss 17214 tgval 20759 mapdunirnN 36939 |
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