Theorem iundifdifd 32488

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 5050	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥)
2		intiin 5035	. . . . . 6 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
3	2	difeq2i 4098	. . . . 5 ⊢ (𝑂 ∖ ∩ 𝐴) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥)
4	1, 3	eqtr4i 2761	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝐴)
5	4	difeq2i 4098	. . 3 ⊢ (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)) = (𝑂 ∖ (𝑂 ∖ ∩ 𝐴))
6		intssuni2 4949	. . . . 5 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝒫 𝑂)
7		unipw 5425	. . . . 5 ⊢ ∪ 𝒫 𝑂 = 𝑂
8	6, 7	sseqtrdi 3999	. . . 4 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ 𝑂)
9		dfss4 4244	. . . 4 ⊢ (∩ 𝐴 ⊆ 𝑂 ↔ (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴)
10	8, 9	sylib 218	. . 3 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴)
11	5, 10	eqtr2id 2783	. 2 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)))
12	11	ex 412	1 ⊢ (𝐴 ⊆ 𝒫 𝑂 → (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ≠ wne 2932   ∖ cdif 3923   ⊆ wss 3926  ∅c0 4308  𝒫 cpw 4575  ∪ cuni 4883  ∩ cint 4922  ∪ ciun 4967  ∩ ciin 4968

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 2707  ax-sep 5266  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-pw 4577  df-sn 4602  df-pr 4604  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970

This theorem is referenced by:  sigaclci  34109