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Theorem iundifdifd 31307
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 𝑥𝐴 (𝑂𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑂

Proof of Theorem iundifdifd
StepHypRef Expression
1 iundif2 5032 . . . . 5 𝑥𝐴 (𝑂𝑥) = (𝑂 𝑥𝐴 𝑥)
2 intiin 5017 . . . . . 6 𝐴 = 𝑥𝐴 𝑥
32difeq2i 4077 . . . . 5 (𝑂 𝐴) = (𝑂 𝑥𝐴 𝑥)
41, 3eqtr4i 2768 . . . 4 𝑥𝐴 (𝑂𝑥) = (𝑂 𝐴)
54difeq2i 4077 . . 3 (𝑂 𝑥𝐴 (𝑂𝑥)) = (𝑂 ∖ (𝑂 𝐴))
6 intssuni2 4932 . . . . 5 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴 𝒫 𝑂)
7 unipw 5405 . . . . 5 𝒫 𝑂 = 𝑂
86, 7sseqtrdi 3992 . . . 4 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴𝑂)
9 dfss4 4216 . . . 4 ( 𝐴𝑂 ↔ (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
108, 9sylib 217 . . 3 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
115, 10eqtr2id 2790 . 2 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))
1211ex 413 1 (𝐴 ⊆ 𝒫 𝑂 → (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wne 2941  cdif 3905  wss 3908  c0 4280  𝒫 cpw 4558   cuni 4863   cint 4905   ciun 4952   ciin 4953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-pw 4560  df-sn 4585  df-pr 4587  df-uni 4864  df-int 4906  df-iun 4954  df-iin 4955
This theorem is referenced by:  sigaclci  32534
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