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Mirrors > Home > NFE Home > Th. List > iinuni | GIF version |
Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
Ref | Expression |
---|---|
iinuni | ⊢ (A ∪ ∩B) = ∩x ∈ B (A ∪ x) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.32v 2757 | . . . 4 ⊢ (∀x ∈ B (y ∈ A ∨ y ∈ x) ↔ (y ∈ A ∨ ∀x ∈ B y ∈ x)) | |
2 | elun 3220 | . . . . 5 ⊢ (y ∈ (A ∪ x) ↔ (y ∈ A ∨ y ∈ x)) | |
3 | 2 | ralbii 2638 | . . . 4 ⊢ (∀x ∈ B y ∈ (A ∪ x) ↔ ∀x ∈ B (y ∈ A ∨ y ∈ x)) |
4 | vex 2862 | . . . . . 6 ⊢ y ∈ V | |
5 | 4 | elint2 3933 | . . . . 5 ⊢ (y ∈ ∩B ↔ ∀x ∈ B y ∈ x) |
6 | 5 | orbi2i 505 | . . . 4 ⊢ ((y ∈ A ∨ y ∈ ∩B) ↔ (y ∈ A ∨ ∀x ∈ B y ∈ x)) |
7 | 1, 3, 6 | 3bitr4ri 269 | . . 3 ⊢ ((y ∈ A ∨ y ∈ ∩B) ↔ ∀x ∈ B y ∈ (A ∪ x)) |
8 | elun 3220 | . . 3 ⊢ (y ∈ (A ∪ ∩B) ↔ (y ∈ A ∨ y ∈ ∩B)) | |
9 | eliin 3974 | . . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ B (A ∪ x) ↔ ∀x ∈ B y ∈ (A ∪ x))) | |
10 | 4, 9 | ax-mp 5 | . . 3 ⊢ (y ∈ ∩x ∈ B (A ∪ x) ↔ ∀x ∈ B y ∈ (A ∪ x)) |
11 | 7, 8, 10 | 3bitr4i 268 | . 2 ⊢ (y ∈ (A ∪ ∩B) ↔ y ∈ ∩x ∈ B (A ∪ x)) |
12 | 11 | eqriv 2350 | 1 ⊢ (A ∪ ∩B) = ∩x ∈ B (A ∪ x) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∨ wo 357 = wceq 1642 ∈ wcel 1710 ∀wral 2614 Vcvv 2859 ∪ cun 3207 ∩cint 3926 ∩ciin 3970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-int 3927 df-iin 3972 |
This theorem is referenced by: (None) |
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