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Mirrors > Home > ILE Home > Th. List > unipr | 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 1559 | . . . 4 | |
2 | vex 2604 | . . . . . . . 8 | |
3 | 2 | elpr 3419 | . . . . . . 7 |
4 | 3 | anbi2i 444 | . . . . . 6 |
5 | andi 764 | . . . . . 6 | |
6 | 4, 5 | bitri 182 | . . . . 5 |
7 | 6 | exbii 1536 | . . . 4 |
8 | unipr.1 | . . . . . . 7 | |
9 | 8 | clel3 2730 | . . . . . 6 |
10 | exancom 1539 | . . . . . 6 | |
11 | 9, 10 | bitri 182 | . . . . 5 |
12 | unipr.2 | . . . . . . 7 | |
13 | 12 | clel3 2730 | . . . . . 6 |
14 | exancom 1539 | . . . . . 6 | |
15 | 13, 14 | bitri 182 | . . . . 5 |
16 | 11, 15 | orbi12i 713 | . . . 4 |
17 | 1, 7, 16 | 3bitr4ri 211 | . . 3 |
18 | 17 | abbii 2194 | . 2 |
19 | df-un 2977 | . 2 | |
20 | df-uni 3602 | . 2 | |
21 | 18, 19, 20 | 3eqtr4ri 2112 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wo 661 wceq 1284 wex 1421 wcel 1433 cab 2067 cvv 2601 cun 2971 cpr 3399 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 |
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-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-uni 3602 |
This theorem is referenced by: uniprg 3616 unisn 3617 uniop 4010 unex 4194 bj-unex 10710 |
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