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Theorem difin0ss 4113
Description: Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
Assertion
Ref Expression
difin0ss (((𝐴𝐵) ∩ 𝐶) = ∅ → (𝐶𝐴𝐶𝐵))

Proof of Theorem difin0ss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eq0 4095 . 2 (((𝐴𝐵) ∩ 𝐶) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶))
2 iman 390 . . . . . 6 ((𝑥𝐶 → (𝑥𝐴𝑥𝐵)) ↔ ¬ (𝑥𝐶 ∧ ¬ (𝑥𝐴𝑥𝐵)))
3 elin 3960 . . . . . . 7 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶))
4 eldif 3744 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
54anbi2ci 618 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶) ↔ (𝑥𝐶 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
6 annim 392 . . . . . . . 8 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
76anbi2i 616 . . . . . . 7 ((𝑥𝐶 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) ↔ (𝑥𝐶 ∧ ¬ (𝑥𝐴𝑥𝐵)))
83, 5, 73bitri 288 . . . . . 6 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ (𝑥𝐶 ∧ ¬ (𝑥𝐴𝑥𝐵)))
92, 8xchbinxr 326 . . . . 5 ((𝑥𝐶 → (𝑥𝐴𝑥𝐵)) ↔ ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶))
10 ax-2 7 . . . . 5 ((𝑥𝐶 → (𝑥𝐴𝑥𝐵)) → ((𝑥𝐶𝑥𝐴) → (𝑥𝐶𝑥𝐵)))
119, 10sylbir 226 . . . 4 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) → ((𝑥𝐶𝑥𝐴) → (𝑥𝐶𝑥𝐵)))
1211al2imi 1910 . . 3 (∀𝑥 ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) → (∀𝑥(𝑥𝐶𝑥𝐴) → ∀𝑥(𝑥𝐶𝑥𝐵)))
13 dfss2 3751 . . 3 (𝐶𝐴 ↔ ∀𝑥(𝑥𝐶𝑥𝐴))
14 dfss2 3751 . . 3 (𝐶𝐵 ↔ ∀𝑥(𝑥𝐶𝑥𝐵))
1512, 13, 143imtr4g 287 . 2 (∀𝑥 ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) → (𝐶𝐴𝐶𝐵))
161, 15sylbi 208 1 (((𝐴𝐵) ∩ 𝐶) = ∅ → (𝐶𝐴𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1650   = wceq 1652  wcel 2155  cdif 3731  cin 3733  wss 3734  c0 4081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-dif 3737  df-in 3741  df-ss 3748  df-nul 4082
This theorem is referenced by:  tz7.7  5936  tfi  7255  lebnumlem3  23057
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