Theorem iundifdif 32651

Description: The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 32650. (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 5003	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥)
2		intiin 4989	. . . . 5 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
3	2	difeq2i 4054	. . . 4 ⊢ (𝑂 ∖ ∩ 𝐴) = (𝑂 ∖ ∩ 𝑥 ∈ 𝐴 𝑥)
4	1, 3	eqtr4i 2765	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥) = (𝑂 ∖ ∩ 𝐴)
5	4	difeq2i 4054	. 2 ⊢ (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)) = (𝑂 ∖ (𝑂 ∖ ∩ 𝐴))
6		iundifdif.2	. . . . 5 ⊢ 𝐴 ⊆ 𝒫 𝑂
7	6	jctl 528	. . . 4 ⊢ (𝐴 ≠ ∅ → (𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅))
8		intssuni2 4903	. . . 4 ⊢ ((𝐴 ⊆ 𝒫 𝑂 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝒫 𝑂)
9		unipw 5389	. . . . . 6 ⊢ ∪ 𝒫 𝑂 = 𝑂
10	9	sseq2i 3944	. . . . 5 ⊢ (∩ 𝐴 ⊆ ∪ 𝒫 𝑂 ↔ ∩ 𝐴 ⊆ 𝑂)
11	10	biimpi 217	. . . 4 ⊢ (∩ 𝐴 ⊆ ∪ 𝒫 𝑂 → ∩ 𝐴 ⊆ 𝑂)
12	7, 8, 11	3syl 18	. . 3 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑂)
13		dfss4 4197	. . 3 ⊢ (∩ 𝐴 ⊆ 𝑂 ↔ (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴)
14	12, 13	sylib 219	. 2 ⊢ (𝐴 ≠ ∅ → (𝑂 ∖ (𝑂 ∖ ∩ 𝐴)) = ∩ 𝐴)
15	5, 14	eqtr2id 2787	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 = (𝑂 ∖ ∪ 𝑥 ∈ 𝐴 (𝑂 ∖ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  Vcvv 3431   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  ∪ cuni 4838  ∩ cint 4877  ∪ ciun 4921  ∩ ciin 4922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-pw 4531  df-sn 4556  df-pr 4558  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924

This theorem is referenced by: (None)