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Mirrors > Home > MPE Home > Th. List > undi | Structured version Visualization version GIF version |
Description: Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
undi | ⊢ (𝐴 ∪ (𝐵 ∩ 𝐶)) = ((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3909 | . . . 4 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) | |
2 | 1 | orbi2i 911 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))) |
3 | ordi 1004 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶))) | |
4 | elin 3909 | . . . 4 ⊢ (𝑥 ∈ ((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑥 ∈ (𝐴 ∪ 𝐶))) | |
5 | elun 4090 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
6 | elun 4090 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐶) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶)) | |
7 | 5, 6 | anbi12i 628 | . . . 4 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑥 ∈ (𝐴 ∪ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶))) |
8 | 4, 7 | bitr2i 277 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶))) |
9 | 2, 3, 8 | 3bitri 298 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶))) |
10 | 9 | uneqri 4092 | 1 ⊢ (𝐴 ∪ (𝐵 ∩ 𝐶)) = ((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∨ wo 845 = wceq 1539 ∈ wcel 2104 ∪ cun 3891 ∩ cin 3892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3440 df-un 3898 df-in 3900 |
This theorem is referenced by: undir 4217 dfif4 4481 dfif5 4482 |
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