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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iundifdifd | Structured version Visualization version GIF version |
Description: The intersection of a set is the complement of the union of the complements. (Contributed by Thierry Arnoux, 19-Dec-2016.) |
Ref | Expression |
---|---|
iundifdifd | ⊢ (𝐴 ⊆ 𝒫 𝑂 → (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iundif2 5032 | . . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥) | |
2 | intiin 5017 | . . . . . 6 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
3 | 2 | difeq2i 4077 | . . . . 5 ⊢ (𝑂 ∖ ∩ 𝐴) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥) |
4 | 1, 3 | eqtr4i 2768 | . . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝐴) |
5 | 4 | difeq2i 4077 | . . 3 ⊢ (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)) = (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) |
6 | intssuni2 4932 | . . . . 5 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝒫 𝑂) | |
7 | unipw 5405 | . . . . 5 ⊢ ∪ 𝒫 𝑂 = 𝑂 | |
8 | 6, 7 | sseqtrdi 3992 | . . . 4 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ 𝑂) |
9 | dfss4 4216 | . . . 4 ⊢ (∩ 𝐴 ⊆ 𝑂 ↔ (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴) | |
10 | 8, 9 | sylib 217 | . . 3 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴) |
11 | 5, 10 | eqtr2id 2790 | . 2 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥))) |
12 | 11 | ex 413 | 1 ⊢ (𝐴 ⊆ 𝒫 𝑂 → (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ≠ wne 2941 ∖ cdif 3905 ⊆ wss 3908 ∅c0 4280 𝒫 cpw 4558 ∪ cuni 4863 ∩ cint 4905 ∪ ciun 4952 ∩ ciin 4953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-sep 5254 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-pw 4560 df-sn 4585 df-pr 4587 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 |
This theorem is referenced by: sigaclci 32534 |
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