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Mirrors > Home > MPE Home > Th. List > iinuni | Structured version Visualization version 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 3342 | . . . 4 ⊢ (∀𝑥 ∈ 𝐵 (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝐴 ∨ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)) | |
2 | elun 4127 | . . . . 5 ⊢ (𝑦 ∈ (𝐴 ∪ 𝑥) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝑥)) | |
3 | 2 | ralbii 3167 | . . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∪ 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝑥)) |
4 | vex 3499 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 4 | elint2 4885 | . . . . 5 ⊢ (𝑦 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥) |
6 | 5 | orbi2i 909 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∨ 𝑦 ∈ ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∨ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)) |
7 | 1, 3, 6 | 3bitr4ri 306 | . . 3 ⊢ ((𝑦 ∈ 𝐴 ∨ 𝑦 ∈ ∩ 𝐵) ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∪ 𝑥)) |
8 | 7 | abbii 2888 | . 2 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ ∩ 𝐵)} = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∪ 𝑥)} |
9 | df-un 3943 | . 2 ⊢ (𝐴 ∪ ∩ 𝐵) = {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ ∩ 𝐵)} | |
10 | df-iin 4924 | . 2 ⊢ ∩ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∪ 𝑥)} | |
11 | 8, 9, 10 | 3eqtr4i 2856 | 1 ⊢ (𝐴 ∪ ∩ 𝐵) = ∩ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 843 = wceq 1537 ∈ wcel 2114 {cab 2801 ∀wral 3140 ∪ cun 3936 ∩ cint 4878 ∩ ciin 4922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-v 3498 df-un 3943 df-int 4879 df-iin 4924 |
This theorem is referenced by: (None) |
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