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Theorem difin0ss 4327
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 4307 . 2 (((𝐴𝐵) ∩ 𝐶) = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶))
2 iman 402 . . . . . 6 ((𝑥𝐶 → (𝑥𝐴𝑥𝐵)) ↔ ¬ (𝑥𝐶 ∧ ¬ (𝑥𝐴𝑥𝐵)))
3 elin 4168 . . . . . . 7 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶))
4 eldif 3945 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
54anbi2ci 624 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶) ↔ (𝑥𝐶 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
6 annim 404 . . . . . . . 8 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
76anbi2i 622 . . . . . . 7 ((𝑥𝐶 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) ↔ (𝑥𝐶 ∧ ¬ (𝑥𝐴𝑥𝐵)))
83, 5, 73bitri 298 . . . . . 6 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ (𝑥𝐶 ∧ ¬ (𝑥𝐴𝑥𝐵)))
92, 8xchbinxr 336 . . . . 5 ((𝑥𝐶 → (𝑥𝐴𝑥𝐵)) ↔ ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶))
10 ax-2 7 . . . . 5 ((𝑥𝐶 → (𝑥𝐴𝑥𝐵)) → ((𝑥𝐶𝑥𝐴) → (𝑥𝐶𝑥𝐵)))
119, 10sylbir 236 . . . 4 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) → ((𝑥𝐶𝑥𝐴) → (𝑥𝐶𝑥𝐵)))
1211al2imi 1807 . . 3 (∀𝑥 ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) → (∀𝑥(𝑥𝐶𝑥𝐴) → ∀𝑥(𝑥𝐶𝑥𝐵)))
13 dfss2 3954 . . 3 (𝐶𝐴 ↔ ∀𝑥(𝑥𝐶𝑥𝐴))
14 dfss2 3954 . . 3 (𝐶𝐵 ↔ ∀𝑥(𝑥𝐶𝑥𝐵))
1512, 13, 143imtr4g 297 . 2 (∀𝑥 ¬ 𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) → (𝐶𝐴𝐶𝐵))
161, 15sylbi 218 1 (((𝐴𝐵) ∩ 𝐶) = ∅ → (𝐶𝐴𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1526   = wceq 1528  wcel 2105  cdif 3932  cin 3934  wss 3935  c0 4290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-dif 3938  df-in 3942  df-ss 3951  df-nul 4291
This theorem is referenced by:  tz7.7  6211  tfi  7556  lebnumlem3  23496
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