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Theorem ssin0 39740
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 3983 . . . 4 ((𝐶𝐴𝐷𝐵) → (𝐶𝐷) ⊆ (𝐴𝐵))
213adant1 1125 . . 3 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐷) ⊆ (𝐴𝐵))
3 eqimss 3798 . . . 4 ((𝐴𝐵) = ∅ → (𝐴𝐵) ⊆ ∅)
433ad2ant1 1128 . . 3 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐴𝐵) ⊆ ∅)
52, 4sstrd 3754 . 2 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐷) ⊆ ∅)
6 ss0 4117 . 2 ((𝐶𝐷) ⊆ ∅ → (𝐶𝐷) = ∅)
75, 6syl 17 1 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐷) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  cin 3714  wss 3715  c0 4058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-in 3722  df-ss 3729  df-nul 4059
This theorem is referenced by:  sge0resplit  41144
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