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Theorem iundifdifd 32574
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 5074 . . . . 5 𝑥𝐴 (𝑂𝑥) = (𝑂 𝑥𝐴 𝑥)
2 intiin 5059 . . . . . 6 𝐴 = 𝑥𝐴 𝑥
32difeq2i 4123 . . . . 5 (𝑂 𝐴) = (𝑂 𝑥𝐴 𝑥)
41, 3eqtr4i 2768 . . . 4 𝑥𝐴 (𝑂𝑥) = (𝑂 𝐴)
54difeq2i 4123 . . 3 (𝑂 𝑥𝐴 (𝑂𝑥)) = (𝑂 ∖ (𝑂 𝐴))
6 intssuni2 4973 . . . . 5 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴 𝒫 𝑂)
7 unipw 5455 . . . . 5 𝒫 𝑂 = 𝑂
86, 7sseqtrdi 4024 . . . 4 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴𝑂)
9 dfss4 4269 . . . 4 ( 𝐴𝑂 ↔ (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
108, 9sylib 218 . . 3 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
115, 10eqtr2id 2790 . 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 1540  wne 2940  cdif 3948  wss 3951  c0 4333  𝒫 cpw 4600   cuni 4907   cint 4946   ciun 4991   ciin 4992
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-pw 4602  df-sn 4627  df-pr 4629  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994
This theorem is referenced by:  sigaclci  34133
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