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Theorem trrelssd 14682
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 14681 . 2 (𝜑 → (𝑆𝑇) ⊆ (𝑅𝑅))
4 trrelssd.r . 2 (𝜑 → (𝑅𝑅) ⊆ 𝑅)
53, 4sstrd 3932 1 (𝜑 → (𝑆𝑇) ⊆ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3888  ccom 5595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3433  df-in 3895  df-ss 3905  df-br 5077  df-opab 5139  df-co 5600
This theorem is referenced by:  trclfvlb2  14719  trrelind  41243  iunrelexpmin1  41286  iunrelexpmin2  41290
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