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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 30003. (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 4895 | . . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥) | |
2 | intiin 4882 | . . . . 5 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
3 | 2 | difeq2i 4017 | . . . 4 ⊢ (𝑂 ∖ ∩ 𝐴) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥) |
4 | 1, 3 | eqtr4i 2822 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝐴) |
5 | 4 | difeq2i 4017 | . 2 ⊢ (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)) = (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) |
6 | iundifdif.2 | . . . . 5 ⊢ 𝐴 ⊆ 𝒫 𝑂 | |
7 | 6 | jctl 524 | . . . 4 ⊢ (𝐴 ≠ ∅ → (𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅)) |
8 | intssuni2 4807 | . . . 4 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝒫 𝑂) | |
9 | unipw 5234 | . . . . . 6 ⊢ ∪ 𝒫 𝑂 = 𝑂 | |
10 | 9 | sseq2i 3917 | . . . . 5 ⊢ (∩ 𝐴 ⊆ ∪ 𝒫 𝑂 ↔ ∩ 𝐴 ⊆ 𝑂) |
11 | 10 | biimpi 217 | . . . 4 ⊢ (∩ 𝐴 ⊆ ∪ 𝒫 𝑂 → ∩ 𝐴 ⊆ 𝑂) |
12 | 7, 8, 11 | 3syl 18 | . . 3 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑂) |
13 | dfss4 4155 | . . 3 ⊢ (∩ 𝐴 ⊆ 𝑂 ↔ (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴) | |
14 | 12, 13 | sylib 219 | . 2 ⊢ (𝐴 ≠ ∅ → (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴) |
15 | 5, 14 | syl5req 2844 | 1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 Vcvv 3437 ∖ cdif 3856 ⊆ wss 3859 ∅c0 4211 𝒫 cpw 4453 ∪ cuni 4745 ∩ cint 4782 ∪ ciun 4825 ∩ ciin 4826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-pw 4455 df-sn 4473 df-pr 4475 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 |
This theorem is referenced by: (None) |
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