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Mirrors > Home > NFE Home > Th. List > iununi | Unicode version |
Description: A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
Ref | Expression |
---|---|
iununi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2518 | . . . . . . 7 | |
2 | iunconst 3977 | . . . . . . 7 | |
3 | 1, 2 | sylbir 204 | . . . . . 6 |
4 | iun0 4022 | . . . . . . 7 | |
5 | id 19 | . . . . . . . 8 | |
6 | 5 | iuneq2d 3994 | . . . . . . 7 |
7 | 4, 6, 5 | 3eqtr4a 2411 | . . . . . 6 |
8 | 3, 7 | ja 153 | . . . . 5 |
9 | 8 | eqcomd 2358 | . . . 4 |
10 | 9 | uneq1d 3417 | . . 3 |
11 | uniiun 4019 | . . . 4 | |
12 | 11 | uneq2i 3415 | . . 3 |
13 | iunun 4046 | . . 3 | |
14 | 10, 12, 13 | 3eqtr4g 2410 | . 2 |
15 | unieq 3900 | . . . . . . 7 | |
16 | uni0 3918 | . . . . . . 7 | |
17 | 15, 16 | syl6eq 2401 | . . . . . 6 |
18 | 17 | uneq2d 3418 | . . . . 5 |
19 | un0 3575 | . . . . 5 | |
20 | 18, 19 | syl6eq 2401 | . . . 4 |
21 | iuneq1 3982 | . . . . 5 | |
22 | 0iun 4023 | . . . . 5 | |
23 | 21, 22 | syl6eq 2401 | . . . 4 |
24 | 20, 23 | eqeq12d 2367 | . . 3 |
25 | 24 | biimpcd 215 | . 2 |
26 | 14, 25 | impbii 180 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 176 wceq 1642 wne 2516 cun 3207 c0 3550 cuni 3891 ciun 3969 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-uni 3892 df-iun 3971 |
This theorem is referenced by: (None) |
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