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Theorem iundifdif 30004
Description: The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 30003. (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 4895 . . . 4 𝑥𝐴 (𝑂𝑥) = (𝑂 𝑥𝐴 𝑥)
2 intiin 4882 . . . . 5 𝐴 = 𝑥𝐴 𝑥
32difeq2i 4017 . . . 4 (𝑂 𝐴) = (𝑂 𝑥𝐴 𝑥)
41, 3eqtr4i 2822 . . 3 𝑥𝐴 (𝑂𝑥) = (𝑂 𝐴)
54difeq2i 4017 . 2 (𝑂 𝑥𝐴 (𝑂𝑥)) = (𝑂 ∖ (𝑂 𝐴))
6 iundifdif.2 . . . . 5 𝐴 ⊆ 𝒫 𝑂
76jctl 524 . . . 4 (𝐴 ≠ ∅ → (𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅))
8 intssuni2 4807 . . . 4 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴 𝒫 𝑂)
9 unipw 5234 . . . . . 6 𝒫 𝑂 = 𝑂
109sseq2i 3917 . . . . 5 ( 𝐴 𝒫 𝑂 𝐴𝑂)
1110biimpi 217 . . . 4 ( 𝐴 𝒫 𝑂 𝐴𝑂)
127, 8, 113syl 18 . . 3 (𝐴 ≠ ∅ → 𝐴𝑂)
13 dfss4 4155 . . 3 ( 𝐴𝑂 ↔ (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
1412, 13sylib 219 . 2 (𝐴 ≠ ∅ → (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
155, 14syl5req 2844 1 (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  wne 2984  Vcvv 3437  cdif 3856  wss 3859  c0 4211  𝒫 cpw 4453   cuni 4745   cint 4782   ciun 4825   ciin 4826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-pw 4455  df-sn 4473  df-pr 4475  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828
This theorem is referenced by: (None)
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