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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 31793. (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 5078 | . . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥) | |
2 | intiin 5063 | . . . . 5 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
3 | 2 | difeq2i 4120 | . . . 4 ⊢ (𝑂 ∖ ∩ 𝐴) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥) |
4 | 1, 3 | eqtr4i 2764 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝐴) |
5 | 4 | difeq2i 4120 | . 2 ⊢ (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)) = (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) |
6 | iundifdif.2 | . . . . 5 ⊢ 𝐴 ⊆ 𝒫 𝑂 | |
7 | 6 | jctl 525 | . . . 4 ⊢ (𝐴 ≠ ∅ → (𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅)) |
8 | intssuni2 4978 | . . . 4 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝒫 𝑂) | |
9 | unipw 5451 | . . . . . 6 ⊢ ∪ 𝒫 𝑂 = 𝑂 | |
10 | 9 | sseq2i 4012 | . . . . 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 4259 | . . 3 ⊢ (∩ 𝐴 ⊆ 𝑂 ↔ (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴) | |
14 | 12, 13 | sylib 217 | . 2 ⊢ (𝐴 ≠ ∅ → (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴) |
15 | 5, 14 | eqtr2id 2786 | 1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ∖ cdif 3946 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 ∪ cuni 4909 ∩ cint 4951 ∪ ciun 4998 ∩ ciin 4999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-pw 4605 df-sn 4630 df-pr 4632 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 |
This theorem is referenced by: (None) |
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