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Theorem difin0ss 3900
 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 3888 . 2 (((𝐴𝐵) ∩ 𝐶) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶))
2 iman 439 . . . . . 6 ((𝑥𝐶 → (𝑥𝐴𝑥𝐵)) ↔ ¬ (𝑥𝐶 ∧ ¬ (𝑥𝐴𝑥𝐵)))
3 elin 3758 . . . . . . 7 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶))
4 eldif 3550 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
54anbi2ci 728 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶) ↔ (𝑥𝐶 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
6 annim 440 . . . . . . . 8 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
76anbi2i 726 . . . . . . 7 ((𝑥𝐶 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) ↔ (𝑥𝐶 ∧ ¬ (𝑥𝐴𝑥𝐵)))
83, 5, 73bitri 285 . . . . . 6 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ (𝑥𝐶 ∧ ¬ (𝑥𝐴𝑥𝐵)))
92, 8xchbinxr 324 . . . . 5 ((𝑥𝐶 → (𝑥𝐴𝑥𝐵)) ↔ ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶))
10 ax-2 7 . . . . 5 ((𝑥𝐶 → (𝑥𝐴𝑥𝐵)) → ((𝑥𝐶𝑥𝐴) → (𝑥𝐶𝑥𝐵)))
119, 10sylbir 224 . . . 4 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) → ((𝑥𝐶𝑥𝐴) → (𝑥𝐶𝑥𝐵)))
1211al2imi 1733 . . 3 (∀𝑥 ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) → (∀𝑥(𝑥𝐶𝑥𝐴) → ∀𝑥(𝑥𝐶𝑥𝐵)))
13 dfss2 3557 . . 3 (𝐶𝐴 ↔ ∀𝑥(𝑥𝐶𝑥𝐴))
14 dfss2 3557 . . 3 (𝐶𝐵 ↔ ∀𝑥(𝑥𝐶𝑥𝐵))
1512, 13, 143imtr4g 284 . 2 (∀𝑥 ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) → (𝐶𝐴𝐶𝐵))
161, 15sylbi 206 1 (((𝐴𝐵) ∩ 𝐶) = ∅ → (𝐶𝐴𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875 This theorem is referenced by:  tz7.7  5666  tfi  6945  lebnumlem3  22570
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