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Theorem iundifdif 32061
Description: The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 32060. (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 𝑥𝐴 (𝑂𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑂

Proof of Theorem iundifdif
StepHypRef Expression
1 iundif2 5076 . . . 4 𝑥𝐴 (𝑂𝑥) = (𝑂 𝑥𝐴 𝑥)
2 intiin 5061 . . . . 5 𝐴 = 𝑥𝐴 𝑥
32difeq2i 4118 . . . 4 (𝑂 𝐴) = (𝑂 𝑥𝐴 𝑥)
41, 3eqtr4i 2761 . . 3 𝑥𝐴 (𝑂𝑥) = (𝑂 𝐴)
54difeq2i 4118 . 2 (𝑂 𝑥𝐴 (𝑂𝑥)) = (𝑂 ∖ (𝑂 𝐴))
6 iundifdif.2 . . . . 5 𝐴 ⊆ 𝒫 𝑂
76jctl 522 . . . 4 (𝐴 ≠ ∅ → (𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅))
8 intssuni2 4976 . . . 4 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴 𝒫 𝑂)
9 unipw 5449 . . . . . 6 𝒫 𝑂 = 𝑂
109sseq2i 4010 . . . . 5 ( 𝐴 𝒫 𝑂 𝐴𝑂)
1110biimpi 215 . . . 4 ( 𝐴 𝒫 𝑂 𝐴𝑂)
127, 8, 113syl 18 . . 3 (𝐴 ≠ ∅ → 𝐴𝑂)
13 dfss4 4257 . . 3 ( 𝐴𝑂 ↔ (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
1412, 13sylib 217 . 2 (𝐴 ≠ ∅ → (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
155, 14eqtr2id 2783 1 (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  wne 2938  Vcvv 3472  cdif 3944  wss 3947  c0 4321  𝒫 cpw 4601   cuni 4907   cint 4949   ciun 4996   ciin 4997
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-pw 4603  df-sn 4628  df-pr 4630  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999
This theorem is referenced by: (None)
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