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Theorem iundifdifd 32581
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 5078 . . . . 5 𝑥𝐴 (𝑂𝑥) = (𝑂 𝑥𝐴 𝑥)
2 intiin 5063 . . . . . 6 𝐴 = 𝑥𝐴 𝑥
32difeq2i 4132 . . . . 5 (𝑂 𝐴) = (𝑂 𝑥𝐴 𝑥)
41, 3eqtr4i 2765 . . . 4 𝑥𝐴 (𝑂𝑥) = (𝑂 𝐴)
54difeq2i 4132 . . 3 (𝑂 𝑥𝐴 (𝑂𝑥)) = (𝑂 ∖ (𝑂 𝐴))
6 intssuni2 4977 . . . . 5 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴 𝒫 𝑂)
7 unipw 5460 . . . . 5 𝒫 𝑂 = 𝑂
86, 7sseqtrdi 4045 . . . 4 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴𝑂)
9 dfss4 4274 . . . 4 ( 𝐴𝑂 ↔ (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
108, 9sylib 218 . . 3 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
115, 10eqtr2id 2787 . 2 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))
1211ex 412 1 (𝐴 ⊆ 𝒫 𝑂 → (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wne 2937  cdif 3959  wss 3962  c0 4338  𝒫 cpw 4604   cuni 4911   cint 4950   ciun 4995   ciin 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-pw 4606  df-sn 4631  df-pr 4633  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998
This theorem is referenced by:  sigaclci  34112
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