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Mirrors > Home > NFE Home > Th. List > iunun | GIF version |
Description: Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
Ref | Expression |
---|---|
iunun | ⊢ ∪x ∈ A (B ∪ C) = (∪x ∈ A B ∪ ∪x ∈ A C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.43 2766 | . . . 4 ⊢ (∃x ∈ A (y ∈ B ∨ y ∈ C) ↔ (∃x ∈ A y ∈ B ∨ ∃x ∈ A y ∈ C)) | |
2 | elun 3220 | . . . . 5 ⊢ (y ∈ (B ∪ C) ↔ (y ∈ B ∨ y ∈ C)) | |
3 | 2 | rexbii 2639 | . . . 4 ⊢ (∃x ∈ A y ∈ (B ∪ C) ↔ ∃x ∈ A (y ∈ B ∨ y ∈ C)) |
4 | eliun 3973 | . . . . 5 ⊢ (y ∈ ∪x ∈ A B ↔ ∃x ∈ A y ∈ B) | |
5 | eliun 3973 | . . . . 5 ⊢ (y ∈ ∪x ∈ A C ↔ ∃x ∈ A y ∈ C) | |
6 | 4, 5 | orbi12i 507 | . . . 4 ⊢ ((y ∈ ∪x ∈ A B ∨ y ∈ ∪x ∈ A C) ↔ (∃x ∈ A y ∈ B ∨ ∃x ∈ A y ∈ C)) |
7 | 1, 3, 6 | 3bitr4i 268 | . . 3 ⊢ (∃x ∈ A y ∈ (B ∪ C) ↔ (y ∈ ∪x ∈ A B ∨ y ∈ ∪x ∈ A C)) |
8 | eliun 3973 | . . 3 ⊢ (y ∈ ∪x ∈ A (B ∪ C) ↔ ∃x ∈ A y ∈ (B ∪ C)) | |
9 | elun 3220 | . . 3 ⊢ (y ∈ (∪x ∈ A B ∪ ∪x ∈ A C) ↔ (y ∈ ∪x ∈ A B ∨ y ∈ ∪x ∈ A C)) | |
10 | 7, 8, 9 | 3bitr4i 268 | . 2 ⊢ (y ∈ ∪x ∈ A (B ∪ C) ↔ y ∈ (∪x ∈ A B ∪ ∪x ∈ A C)) |
11 | 10 | eqriv 2350 | 1 ⊢ ∪x ∈ A (B ∪ C) = (∪x ∈ A B ∪ ∪x ∈ A C) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 357 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 ∪ cun 3207 ∪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-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-iun 3971 |
This theorem is referenced by: iununi 4050 |
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