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Theorem ssin0 38248
 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 simp2 1055 . . . 4 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → 𝐶𝐴)
2 simp3 1056 . . . 4 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → 𝐷𝐵)
3 ss2in 3802 . . . 4 ((𝐶𝐴𝐷𝐵) → (𝐶𝐷) ⊆ (𝐴𝐵))
41, 2, 3syl2anc 691 . . 3 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐷) ⊆ (𝐴𝐵))
5 eqimss 3620 . . . 4 ((𝐴𝐵) = ∅ → (𝐴𝐵) ⊆ ∅)
653ad2ant1 1075 . . 3 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐴𝐵) ⊆ ∅)
74, 6sstrd 3578 . 2 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐷) ⊆ ∅)
8 ss0 3926 . 2 ((𝐶𝐷) ⊆ ∅ → (𝐶𝐷) = ∅)
97, 8syl 17 1 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐷) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∩ 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-3an 1033  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:  sge0resplit  39299
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