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Mirrors > Home > NFE Home > Th. List > iunxun | GIF version |
Description: Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
Ref | Expression |
---|---|
iunxun | ⊢ ∪x ∈ (A ∪ B)C = (∪x ∈ A C ∪ ∪x ∈ B C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexun 3443 | . . . 4 ⊢ (∃x ∈ (A ∪ B)y ∈ C ↔ (∃x ∈ A y ∈ C ∨ ∃x ∈ B y ∈ C)) | |
2 | eliun 3973 | . . . . 5 ⊢ (y ∈ ∪x ∈ A C ↔ ∃x ∈ A y ∈ C) | |
3 | eliun 3973 | . . . . 5 ⊢ (y ∈ ∪x ∈ B C ↔ ∃x ∈ B y ∈ C) | |
4 | 2, 3 | orbi12i 507 | . . . 4 ⊢ ((y ∈ ∪x ∈ A C ∨ y ∈ ∪x ∈ B C) ↔ (∃x ∈ A y ∈ C ∨ ∃x ∈ B y ∈ C)) |
5 | 1, 4 | bitr4i 243 | . . 3 ⊢ (∃x ∈ (A ∪ B)y ∈ C ↔ (y ∈ ∪x ∈ A C ∨ y ∈ ∪x ∈ B C)) |
6 | eliun 3973 | . . 3 ⊢ (y ∈ ∪x ∈ (A ∪ B)C ↔ ∃x ∈ (A ∪ B)y ∈ C) | |
7 | elun 3220 | . . 3 ⊢ (y ∈ (∪x ∈ A C ∪ ∪x ∈ B C) ↔ (y ∈ ∪x ∈ A C ∨ y ∈ ∪x ∈ B C)) | |
8 | 5, 6, 7 | 3bitr4i 268 | . 2 ⊢ (y ∈ ∪x ∈ (A ∪ B)C ↔ y ∈ (∪x ∈ A C ∪ ∪x ∈ B C)) |
9 | 8 | eqriv 2350 | 1 ⊢ ∪x ∈ (A ∪ B)C = (∪x ∈ A C ∪ ∪x ∈ B 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-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-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-iun 3971 |
This theorem is referenced by: (None) |
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