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Mirrors > Home > MPE Home > Th. List > uniun | Structured version Visualization version GIF version |
Description: The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.) |
Ref | Expression |
---|---|
uniun | ⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.43 1890 | . . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))) | |
2 | elun 4049 | . . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∪ 𝐵) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)) | |
3 | 2 | anbi2i 626 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵))) |
4 | andi 1008 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))) | |
5 | 3, 4 | bitri 278 | . . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))) |
6 | 5 | exbii 1855 | . . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ ∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))) |
7 | eluni 4808 | . . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) | |
8 | eluni 4808 | . . . . 5 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)) | |
9 | 7, 8 | orbi12i 915 | . . . 4 ⊢ ((𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))) |
10 | 1, 6, 9 | 3bitr4i 306 | . . 3 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵)) |
11 | eluni 4808 | . . 3 ⊢ (𝑥 ∈ ∪ (𝐴 ∪ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵))) | |
12 | elun 4049 | . . 3 ⊢ (𝑥 ∈ (∪ 𝐴 ∪ ∪ 𝐵) ↔ (𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵)) | |
13 | 10, 11, 12 | 3bitr4i 306 | . 2 ⊢ (𝑥 ∈ ∪ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (∪ 𝐴 ∪ ∪ 𝐵)) |
14 | 13 | eqriv 2733 | 1 ⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∨ wo 847 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ∪ cun 3851 ∪ cuni 4805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-un 3858 df-uni 4806 |
This theorem is referenced by: unidif0 5236 unisuc 6267 fvssunirn 6724 fvun 6779 onuninsuci 7597 tc2 9336 fin1a2lem10 9988 fin1a2lem12 9990 incexclem 15363 dprd2da 19383 dmdprdsplit2lem 19386 ordtuni 22041 cmpcld 22253 uncmp 22254 refun0 22366 lfinun 22376 1stckgenlem 22404 filconn 22734 ufildr 22782 alexsubALTlem3 22900 cldsubg 22962 icccmplem2 23674 uniioombllem3 24436 sxbrsigalem0 31904 fiunelcarsg 31949 carsgclctunlem1 31950 carsggect 31951 cvmscld 32902 madeoldsuc 33753 refssfne 34233 topjoin 34240 pibt2 35274 mbfresfi 35509 fourierdlem80 43345 isomenndlem 43686 |
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