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Mirrors > Home > MPE Home > Th. List > unipr | Structured version Visualization version Unicode version |
Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.) |
Ref | Expression |
---|---|
unipr.1 | |
unipr.2 |
Ref | Expression |
---|---|
unipr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.43 1810 | . . . 4 | |
2 | vex 3203 | . . . . . . . 8 | |
3 | 2 | elpr 4198 | . . . . . . 7 |
4 | 3 | anbi2i 730 | . . . . . 6 |
5 | andi 911 | . . . . . 6 | |
6 | 4, 5 | bitri 264 | . . . . 5 |
7 | 6 | exbii 1774 | . . . 4 |
8 | unipr.1 | . . . . . . 7 | |
9 | 8 | clel3 3341 | . . . . . 6 |
10 | exancom 1787 | . . . . . 6 | |
11 | 9, 10 | bitri 264 | . . . . 5 |
12 | unipr.2 | . . . . . . 7 | |
13 | 12 | clel3 3341 | . . . . . 6 |
14 | exancom 1787 | . . . . . 6 | |
15 | 13, 14 | bitri 264 | . . . . 5 |
16 | 11, 15 | orbi12i 543 | . . . 4 |
17 | 1, 7, 16 | 3bitr4ri 293 | . . 3 |
18 | 17 | abbii 2739 | . 2 |
19 | df-un 3579 | . 2 | |
20 | df-uni 4437 | . 2 | |
21 | 18, 19, 20 | 3eqtr4ri 2655 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wo 383 wa 384 wceq 1483 wex 1704 wcel 1990 cab 2608 cvv 3200 cun 3572 cpr 4179 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-un 3579 df-sn 4178 df-pr 4180 df-uni 4437 |
This theorem is referenced by: uniprg 4450 unisn 4451 uniintsn 4514 uniop 4977 unex 6956 rankxplim 8742 mrcun 16282 indistps 20815 indistps2 20816 leordtval2 21016 ex-uni 27283 fouriersw 40448 |
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