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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iundifdif | Structured version Visualization version GIF version |
Description: The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 32060. (Contributed by Thierry Arnoux, 4-Sep-2016.) |
Ref | Expression |
---|---|
iundifdif.o | ⊢ 𝑂 ∈ V |
iundifdif.2 | ⊢ 𝐴 ⊆ 𝒫 𝑂 |
Ref | Expression |
---|---|
iundifdif | ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iundif2 5076 | . . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥) | |
2 | intiin 5061 | . . . . 5 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
3 | 2 | difeq2i 4118 | . . . 4 ⊢ (𝑂 ∖ ∩ 𝐴) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥) |
4 | 1, 3 | eqtr4i 2761 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝐴) |
5 | 4 | difeq2i 4118 | . 2 ⊢ (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)) = (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) |
6 | iundifdif.2 | . . . . 5 ⊢ 𝐴 ⊆ 𝒫 𝑂 | |
7 | 6 | jctl 522 | . . . 4 ⊢ (𝐴 ≠ ∅ → (𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅)) |
8 | intssuni2 4976 | . . . 4 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝒫 𝑂) | |
9 | unipw 5449 | . . . . . 6 ⊢ ∪ 𝒫 𝑂 = 𝑂 | |
10 | 9 | sseq2i 4010 | . . . . 5 ⊢ (∩ 𝐴 ⊆ ∪ 𝒫 𝑂 ↔ ∩ 𝐴 ⊆ 𝑂) |
11 | 10 | biimpi 215 | . . . 4 ⊢ (∩ 𝐴 ⊆ ∪ 𝒫 𝑂 → ∩ 𝐴 ⊆ 𝑂) |
12 | 7, 8, 11 | 3syl 18 | . . 3 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑂) |
13 | dfss4 4257 | . . 3 ⊢ (∩ 𝐴 ⊆ 𝑂 ↔ (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴) | |
14 | 12, 13 | sylib 217 | . 2 ⊢ (𝐴 ≠ ∅ → (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴) |
15 | 5, 14 | eqtr2id 2783 | 1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 Vcvv 3472 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 ∪ cuni 4907 ∩ cint 4949 ∪ ciun 4996 ∩ ciin 4997 |
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 2701 ax-sep 5298 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-pw 4603 df-sn 4628 df-pr 4630 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 |
This theorem is referenced by: (None) |
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