Theorem iundifdifd 30416

Description: The intersection of a set is the complement of the union of the complements. (Contributed by Thierry Arnoux, 19-Dec-2016.)

Assertion

Ref	Expression
iundifdifd	⊢ (𝐴 ⊆ 𝒫 𝑂 → (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑂

Proof of Theorem iundifdifd

Step	Hyp	Ref	Expression
1		iundif2 4962	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥)
2		intiin 4949	. . . . . 6 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
3	2	difeq2i 4026	. . . . 5 ⊢ (𝑂 ∖ ∩ 𝐴) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥)
4	1, 3	eqtr4i 2785	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝐴)
5	4	difeq2i 4026	. . 3 ⊢ (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)) = (𝑂 ∖ (𝑂 ∖ ∩ 𝐴))
6		intssuni2 4864	. . . . 5 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝒫 𝑂)
7		unipw 5312	. . . . 5 ⊢ ∪ 𝒫 𝑂 = 𝑂
8	6, 7	sseqtrdi 3943	. . . 4 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ 𝑂)
9		dfss4 4164	. . . 4 ⊢ (∩ 𝐴 ⊆ 𝑂 ↔ (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴)
10	8, 9	sylib 221	. . 3 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴)
11	5, 10	syl5req 2807	. 2 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)))
12	11	ex 417	1 ⊢ (𝐴 ⊆ 𝒫 𝑂 → (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ≠ wne 2952   ∖ cdif 3856   ⊆ wss 3859  ∅c0 4226  𝒫 cpw 4495  ∪ cuni 4799  ∩ cint 4839  ∪ ciun 4884  ∩ ciin 4885

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-pw 4497  df-sn 4524  df-pr 4526  df-uni 4800  df-int 4840  df-iun 4886  df-iin 4887

This theorem is referenced by:  sigaclci  31612