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Theorem iundifdifd 30241
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 4987 . . . . 5 𝑥𝐴 (𝑂𝑥) = (𝑂 𝑥𝐴 𝑥)
2 intiin 4974 . . . . . 6 𝐴 = 𝑥𝐴 𝑥
32difeq2i 4093 . . . . 5 (𝑂 𝐴) = (𝑂 𝑥𝐴 𝑥)
41, 3eqtr4i 2844 . . . 4 𝑥𝐴 (𝑂𝑥) = (𝑂 𝐴)
54difeq2i 4093 . . 3 (𝑂 𝑥𝐴 (𝑂𝑥)) = (𝑂 ∖ (𝑂 𝐴))
6 intssuni2 4892 . . . . 5 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴 𝒫 𝑂)
7 unipw 5333 . . . . 5 𝒫 𝑂 = 𝑂
86, 7sseqtrdi 4014 . . . 4 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴𝑂)
9 dfss4 4232 . . . 4 ( 𝐴𝑂 ↔ (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
108, 9sylib 219 . . 3 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
115, 10syl5req 2866 . 2 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))
1211ex 413 1 (𝐴 ⊆ 𝒫 𝑂 → (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wne 3013  cdif 3930  wss 3933  c0 4288  𝒫 cpw 4535   cuni 4830   cint 4867   ciun 4910   ciin 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-pw 4537  df-sn 4558  df-pr 4560  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913
This theorem is referenced by:  sigaclci  31290
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