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Theorem difin0ss 4301
 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 4281 . 2 (((𝐴𝐵) ∩ 𝐶) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶))
2 iman 405 . . . . . 6 ((𝑥𝐶 → (𝑥𝐴𝑥𝐵)) ↔ ¬ (𝑥𝐶 ∧ ¬ (𝑥𝐴𝑥𝐵)))
3 elin 3926 . . . . . . 7 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶))
4 eldif 3920 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
54anbi2ci 627 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶) ↔ (𝑥𝐶 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
6 annim 407 . . . . . . . 8 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
76anbi2i 625 . . . . . . 7 ((𝑥𝐶 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) ↔ (𝑥𝐶 ∧ ¬ (𝑥𝐴𝑥𝐵)))
83, 5, 73bitri 300 . . . . . 6 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ (𝑥𝐶 ∧ ¬ (𝑥𝐴𝑥𝐵)))
92, 8xchbinxr 338 . . . . 5 ((𝑥𝐶 → (𝑥𝐴𝑥𝐵)) ↔ ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶))
10 ax-2 7 . . . . 5 ((𝑥𝐶 → (𝑥𝐴𝑥𝐵)) → ((𝑥𝐶𝑥𝐴) → (𝑥𝐶𝑥𝐵)))
119, 10sylbir 238 . . . 4 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) → ((𝑥𝐶𝑥𝐴) → (𝑥𝐶𝑥𝐵)))
1211al2imi 1817 . . 3 (∀𝑥 ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) → (∀𝑥(𝑥𝐶𝑥𝐴) → ∀𝑥(𝑥𝐶𝑥𝐵)))
13 dfss2 3930 . . 3 (𝐶𝐴 ↔ ∀𝑥(𝑥𝐶𝑥𝐴))
14 dfss2 3930 . . 3 (𝐶𝐵 ↔ ∀𝑥(𝑥𝐶𝑥𝐵))
1512, 13, 143imtr4g 299 . 2 (∀𝑥 ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) → (𝐶𝐴𝐶𝐵))
161, 15sylbi 220 1 (((𝐴𝐵) ∩ 𝐶) = ∅ → (𝐶𝐴𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2115   ∖ cdif 3907   ∩ cin 3909   ⊆ wss 3910  ∅c0 4266 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-11 2162  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-v 3473  df-dif 3913  df-in 3917  df-ss 3927  df-nul 4267 This theorem is referenced by:  tz7.7  6190  tfi  7543  lebnumlem3  23546
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