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Theorem iundifdif 29706
Description: The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 29705. (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 4779 . . . 4 𝑥𝐴 (𝑂𝑥) = (𝑂 𝑥𝐴 𝑥)
2 intiin 4766 . . . . 5 𝐴 = 𝑥𝐴 𝑥
32difeq2i 3924 . . . 4 (𝑂 𝐴) = (𝑂 𝑥𝐴 𝑥)
41, 3eqtr4i 2831 . . 3 𝑥𝐴 (𝑂𝑥) = (𝑂 𝐴)
54difeq2i 3924 . 2 (𝑂 𝑥𝐴 (𝑂𝑥)) = (𝑂 ∖ (𝑂 𝐴))
6 iundifdif.2 . . . . 5 𝐴 ⊆ 𝒫 𝑂
76jctl 515 . . . 4 (𝐴 ≠ ∅ → (𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅))
8 intssuni2 4694 . . . 4 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴 𝒫 𝑂)
9 unipw 5108 . . . . . 6 𝒫 𝑂 = 𝑂
109sseq2i 3827 . . . . 5 ( 𝐴 𝒫 𝑂 𝐴𝑂)
1110biimpi 207 . . . 4 ( 𝐴 𝒫 𝑂 𝐴𝑂)
127, 8, 113syl 18 . . 3 (𝐴 ≠ ∅ → 𝐴𝑂)
13 dfss4 4060 . . 3 ( 𝐴𝑂 ↔ (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
1412, 13sylib 209 . 2 (𝐴 ≠ ∅ → (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
155, 14syl5req 2853 1 (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  wne 2978  Vcvv 3391  cdif 3766  wss 3769  c0 4116  𝒫 cpw 4351   cuni 4630   cint 4669   ciun 4712   ciin 4713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-pw 4353  df-sn 4371  df-pr 4373  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715
This theorem is referenced by: (None)
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