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Theorem reldisj 4163
Description: Two ways of saying that two classes are disjoint, using the complement of 𝐵 relative to a universe 𝐶. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
reldisj (𝐴𝐶 → ((𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶𝐵)))

Proof of Theorem reldisj
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3740 . . . 4 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
2 pm5.44 532 . . . . . 6 ((𝑥𝐴𝑥𝐶) → ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵))))
3 eldif 3733 . . . . . . 7 (𝑥 ∈ (𝐶𝐵) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐵))
43imbi2i 325 . . . . . 6 ((𝑥𝐴𝑥 ∈ (𝐶𝐵)) ↔ (𝑥𝐴 → (𝑥𝐶 ∧ ¬ 𝑥𝐵)))
52, 4syl6bbr 278 . . . . 5 ((𝑥𝐴𝑥𝐶) → ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐴𝑥 ∈ (𝐶𝐵))))
65sps 2209 . . . 4 (∀𝑥(𝑥𝐴𝑥𝐶) → ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐴𝑥 ∈ (𝐶𝐵))))
71, 6sylbi 207 . . 3 (𝐴𝐶 → ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ (𝑥𝐴𝑥 ∈ (𝐶𝐵))))
87albidv 2001 . 2 (𝐴𝐶 → (∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐶𝐵))))
9 disj1 4162 . 2 ((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
10 dfss2 3740 . 2 (𝐴 ⊆ (𝐶𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐶𝐵)))
118, 9, 103bitr4g 303 1 (𝐴𝐶 → ((𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wcel 2145  cdif 3720  cin 3722  wss 3723  c0 4063
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
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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-ral 3066  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-nul 4064
This theorem is referenced by:  disj2  4168  ssdifsn  4454  oacomf1olem  7798  domdifsn  8199  elfiun  8492  cantnfp1lem3  8741  ssxr  10309  structcnvcnv  16078  fidomndrng  19522  elcls  21098  ist1-2  21372  nrmsep2  21381  nrmsep  21382  isnrm3  21384  isreg2  21402  hauscmplem  21430  connsub  21445  iunconnlem  21451  llycmpkgen2  21574  hausdiag  21669  trfil3  21912  isufil2  21932  filufint  21944  blcld  22530  i1fima2  23666  i1fd  23668  nbgrssvwo2  26481  nbgrssvwo2OLD  26484  pliguhgr  27680  poimirlem15  33757  itg2addnclem2  33794  ntrk0kbimka  38863  ntrneicls11  38914  gneispace  38958
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