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Mirrors > Home > MPE Home > Th. List > iunun | Structured version Visualization version 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 3351 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∨ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) | |
2 | elun 4125 | . . . . 5 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶)) | |
3 | 2 | rexbii 3247 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶)) |
4 | eliun 4916 | . . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
5 | eliun 4916 | . . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) | |
6 | 4, 5 | orbi12i 911 | . . . 4 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∨ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) |
7 | 1, 3, 6 | 3bitr4i 305 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)) |
8 | eliun 4916 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶)) | |
9 | elun 4125 | . . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)) | |
10 | 7, 8, 9 | 3bitr4i 305 | . 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ 𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ 𝐴 𝐶)) |
11 | 10 | eqriv 2818 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 843 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ∪ cun 3934 ∪ ciun 4912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3497 df-un 3941 df-iun 4914 |
This theorem is referenced by: iununi 5014 oarec 8182 comppfsc 22134 uniiccdif 24173 bnj1415 32305 dftrpred4g 33068 |
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