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Theorem iundifdifd 29718
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 4721 . . . . 5 𝑥𝐴 (𝑂𝑥) = (𝑂 𝑥𝐴 𝑥)
2 intiin 4708 . . . . . 6 𝐴 = 𝑥𝐴 𝑥
32difeq2i 3876 . . . . 5 (𝑂 𝐴) = (𝑂 𝑥𝐴 𝑥)
41, 3eqtr4i 2796 . . . 4 𝑥𝐴 (𝑂𝑥) = (𝑂 𝐴)
54difeq2i 3876 . . 3 (𝑂 𝑥𝐴 (𝑂𝑥)) = (𝑂 ∖ (𝑂 𝐴))
6 intssuni2 4636 . . . . 5 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴 𝒫 𝑂)
7 unipw 5046 . . . . 5 𝒫 𝑂 = 𝑂
86, 7syl6sseq 3800 . . . 4 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴𝑂)
9 dfss4 4007 . . . 4 ( 𝐴𝑂 ↔ (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
108, 9sylib 208 . . 3 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → (𝑂 ∖ (𝑂 𝐴)) = 𝐴)
115, 10syl5req 2818 . 2 ((𝐴 ⊆ 𝒫 𝑂𝐴 ≠ ∅) → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))
1211ex 397 1 (𝐴 ⊆ 𝒫 𝑂 → (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wne 2943  cdif 3720  wss 3723  c0 4063  𝒫 cpw 4297   cuni 4574   cint 4611   ciun 4654   ciin 4655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-pw 4299  df-sn 4317  df-pr 4319  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657
This theorem is referenced by:  sigaclci  30535
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