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Mirrors > Home > MPE Home > Th. List > iinun2 | Structured version Visualization version GIF version |
Description: Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4968 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.) |
Ref | Expression |
---|---|
iinun2 | ⊢ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.32v 3254 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) | |
2 | elun 4063 | . . . . 5 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶)) | |
3 | 2 | ralbii 3088 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶)) |
4 | eliin 4909 | . . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) | |
5 | 4 | elv 3414 | . . . . 5 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) |
6 | 5 | orbi2i 913 | . . . 4 ⊢ ((𝑦 ∈ 𝐵 ∨ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) |
7 | 1, 3, 6 | 3bitr4i 306 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)) |
8 | eliin 4909 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶))) | |
9 | 8 | elv 3414 | . . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶)) |
10 | elun 4063 | . . 3 ⊢ (𝑦 ∈ (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)) | |
11 | 7, 9, 10 | 3bitr4i 306 | . 2 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ 𝑦 ∈ (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶)) |
12 | 11 | eqriv 2734 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∨ wo 847 = wceq 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 ∪ cun 3864 ∩ ciin 4905 |
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 2016 ax-8 2112 ax-9 2120 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 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-v 3410 df-un 3871 df-iin 4907 |
This theorem is referenced by: (None) |
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