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Theorem iundifdif 32575
Description: The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 32574. (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 5074 . . . 4 𝑥𝐴 (𝑂𝑥) = (𝑂 𝑥𝐴 𝑥)
2 intiin 5059 . . . . 5 𝐴 = 𝑥𝐴 𝑥
32difeq2i 4123 . . . 4 (𝑂 𝐴) = (𝑂 𝑥𝐴 𝑥)
41, 3eqtr4i 2768 . . 3 𝑥𝐴 (𝑂𝑥) = (𝑂 𝐴)
54difeq2i 4123 . 2 (𝑂 𝑥𝐴 (𝑂𝑥)) = (𝑂 ∖ (𝑂 𝐴))
6 iundifdif.2 . . . . 5 𝐴 ⊆ 𝒫 𝑂
76jctl 523 . . . 4 (𝐴 ≠ ∅ → (𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅))
8 intssuni2 4973 . . . 4 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴 𝒫 𝑂)
9 unipw 5455 . . . . . 6 𝒫 𝑂 = 𝑂
109sseq2i 4013 . . . . 5 ( 𝐴 𝒫 𝑂 𝐴𝑂)
1110biimpi 216 . . . 4 ( 𝐴 𝒫 𝑂 𝐴𝑂)
127, 8, 113syl 18 . . 3 (𝐴 ≠ ∅ → 𝐴𝑂)
13 dfss4 4269 . . 3 ( 𝐴𝑂 ↔ (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
1412, 13sylib 218 . 2 (𝐴 ≠ ∅ → (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
155, 14eqtr2id 2790 1 (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  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: (None)
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