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Theorem ssin0 39537
Description: If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
ssin0 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐷) = ∅)

Proof of Theorem ssin0
StepHypRef Expression
1 ss2in 3873 . . . 4 ((𝐶𝐴𝐷𝐵) → (𝐶𝐷) ⊆ (𝐴𝐵))
213adant1 1099 . . 3 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐷) ⊆ (𝐴𝐵))
3 eqimss 3690 . . . 4 ((𝐴𝐵) = ∅ → (𝐴𝐵) ⊆ ∅)
433ad2ant1 1102 . . 3 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐴𝐵) ⊆ ∅)
52, 4sstrd 3646 . 2 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐷) ⊆ ∅)
6 ss0 4007 . 2 ((𝐶𝐷) ⊆ ∅ → (𝐶𝐷) = ∅)
75, 6syl 17 1 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐷) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054   = wceq 1523  cin 3606  wss 3607  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949
This theorem is referenced by:  sge0resplit  40941
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