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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 32574. (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 5074 | . . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥) | |
| 2 | intiin 5059 | . . . . 5 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
| 3 | 2 | difeq2i 4123 | . . . 4 ⊢ (𝑂 ∖ ∩ 𝐴) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥) |
| 4 | 1, 3 | eqtr4i 2768 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝐴) |
| 5 | 4 | difeq2i 4123 | . 2 ⊢ (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)) = (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) |
| 6 | iundifdif.2 | . . . . 5 ⊢ 𝐴 ⊆ 𝒫 𝑂 | |
| 7 | 6 | jctl 523 | . . . 4 ⊢ (𝐴 ≠ ∅ → (𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅)) |
| 8 | intssuni2 4973 | . . . 4 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝒫 𝑂) | |
| 9 | unipw 5455 | . . . . . 6 ⊢ ∪ 𝒫 𝑂 = 𝑂 | |
| 10 | 9 | sseq2i 4013 | . . . . 5 ⊢ (∩ 𝐴 ⊆ ∪ 𝒫 𝑂 ↔ ∩ 𝐴 ⊆ 𝑂) |
| 11 | 10 | biimpi 216 | . . . 4 ⊢ (∩ 𝐴 ⊆ ∪ 𝒫 𝑂 → ∩ 𝐴 ⊆ 𝑂) |
| 12 | 7, 8, 11 | 3syl 18 | . . 3 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑂) |
| 13 | dfss4 4269 | . . 3 ⊢ (∩ 𝐴 ⊆ 𝑂 ↔ (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴) | |
| 14 | 12, 13 | sylib 218 | . 2 ⊢ (𝐴 ≠ ∅ → (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴) |
| 15 | 5, 14 | eqtr2id 2790 | 1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 ∪ cuni 4907 ∩ cint 4946 ∪ ciun 4991 ∩ ciin 4992 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-pr 4629 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 |
| This theorem is referenced by: (None) |
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