Theorem iundifdif 32506

Description: The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 32505. (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 5023	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥)
2		intiin 5008	. . . . 5 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
3	2	difeq2i 4074	. . . 4 ⊢ (𝑂 ∖ ∩ 𝐴) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥)
4	1, 3	eqtr4i 2755	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝐴)
5	4	difeq2i 4074	. 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 4923	. . . 4 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝒫 𝑂)
9		unipw 5393	. . . . . 6 ⊢ ∪ 𝒫 𝑂 = 𝑂
10	9	sseq2i 3965	. . . . 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 4220	. . 3 ⊢ (∩ 𝐴 ⊆ 𝑂 ↔ (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴)
14	12, 13	sylib 218	. 2 ⊢ (𝐴 ≠ ∅ → (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴)
15	5, 14	eqtr2id 2777	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  ∪ cuni 4858  ∩ cint 4896  ∪ ciun 4941  ∩ ciin 4942

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-pr 5371

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-pw 4553  df-sn 4578  df-pr 4580  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944

This theorem is referenced by: (None)