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Theorem trrelssd 14944
Description: The composition of subclasses of a transitive relation is a subclass of that relation. (Contributed by RP, 24-Dec-2019.)
Hypotheses
Ref Expression
trrelssd.r (𝜑 → (𝑅𝑅) ⊆ 𝑅)
trrelssd.s (𝜑𝑆𝑅)
trrelssd.t (𝜑𝑇𝑅)
Assertion
Ref Expression
trrelssd (𝜑 → (𝑆𝑇) ⊆ 𝑅)

Proof of Theorem trrelssd
StepHypRef Expression
1 trrelssd.s . . 3 (𝜑𝑆𝑅)
2 trrelssd.t . . 3 (𝜑𝑇𝑅)
31, 2coss12d 14943 . 2 (𝜑 → (𝑆𝑇) ⊆ (𝑅𝑅))
4 trrelssd.r . 2 (𝜑 → (𝑅𝑅) ⊆ 𝑅)
53, 4sstrd 3988 1 (𝜑 → (𝑆𝑇) ⊆ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3944  ccom 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3471  df-in 3951  df-ss 3961  df-br 5143  df-opab 5205  df-co 5681
This theorem is referenced by:  trclfvlb2  14981  trrelind  43018  iunrelexpmin1  43061  iunrelexpmin2  43065
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