Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iundifdif Structured version   Visualization version   GIF version

Theorem iundifdif 28763
 Description: The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 28762. (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 4523 . . . 4 𝑥𝐴 (𝑂𝑥) = (𝑂 𝑥𝐴 𝑥)
2 intiin 4510 . . . . 5 𝐴 = 𝑥𝐴 𝑥
32difeq2i 3687 . . . 4 (𝑂 𝐴) = (𝑂 𝑥𝐴 𝑥)
41, 3eqtr4i 2635 . . 3 𝑥𝐴 (𝑂𝑥) = (𝑂 𝐴)
54difeq2i 3687 . 2 (𝑂 𝑥𝐴 (𝑂𝑥)) = (𝑂 ∖ (𝑂 𝐴))
6 iundifdif.2 . . . . 5 𝐴 ⊆ 𝒫 𝑂
76jctl 562 . . . 4 (𝐴 ≠ ∅ → (𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅))
8 intssuni2 4437 . . . 4 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴 𝒫 𝑂)
9 unipw 4845 . . . . . 6 𝒫 𝑂 = 𝑂
109sseq2i 3593 . . . . 5 ( 𝐴 𝒫 𝑂 𝐴𝑂)
1110biimpi 205 . . . 4 ( 𝐴 𝒫 𝑂 𝐴𝑂)
127, 8, 113syl 18 . . 3 (𝐴 ≠ ∅ → 𝐴𝑂)
13 dfss4 3820 . . 3 ( 𝐴𝑂 ↔ (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
1412, 13sylib 207 . 2 (𝐴 ≠ ∅ → (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
155, 14syl5req 2657 1 (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  ∩ cint 4410  ∪ ciun 4455  ∩ ciin 4456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator