Theorem iundifdif 28763

Description: The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 28762. (Contributed by Thierry Arnoux, 4-Sep-2016.)

Hypotheses

Ref	Expression
iundifdif.o	⊢ 𝑂 ∈ V
iundifdif.2	⊢ 𝐴 ⊆ 𝒫 𝑂

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝑂

Proof of Theorem iundifdif

Step	Hyp	Ref	Expression
1		iundif2 4523	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥)
2		intiin 4510	. . . . 5 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
3	2	difeq2i 3687	. . . 4 ⊢ (𝑂 ∖ ∩ 𝐴) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥)
4	1, 3	eqtr4i 2635	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝐴)
5	4	difeq2i 3687	. 2 ⊢ (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)) = (𝑂 ∖ (𝑂 ∖ ∩ 𝐴))
6		iundifdif.2	. . . . 5 ⊢ 𝐴 ⊆ 𝒫 𝑂
7	6	jctl 562	. . . 4 ⊢ (𝐴 ≠ ∅ → (𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅))
8		intssuni2 4437	. . . 4 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝒫 𝑂)
9		unipw 4845	. . . . . 6 ⊢ ∪ 𝒫 𝑂 = 𝑂
10	9	sseq2i 3593	. . . . 5 ⊢ (∩ 𝐴 ⊆ ∪ 𝒫 𝑂 ↔ ∩ 𝐴 ⊆ 𝑂)
11	10	biimpi 205	. . . 4 ⊢ (∩ 𝐴 ⊆ ∪ 𝒫 𝑂 → ∩ 𝐴 ⊆ 𝑂)
12	7, 8, 11	3syl 18	. . 3 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑂)
13		dfss4 3820	. . 3 ⊢ (∩ 𝐴 ⊆ 𝑂 ↔ (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴)
14	12, 13	sylib 207	. 2 ⊢ (𝐴 ≠ ∅ → (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴)
15	5, 14	syl5req 2657	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  ∩ cint 4410  ∪ ciun 4455  ∩ ciin 4456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458

This theorem is referenced by: (None)