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Theorem trrelssd 13934
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 13933 . 2 (𝜑 → (𝑆𝑇) ⊆ (𝑅𝑅))
4 trrelssd.r . 2 (𝜑 → (𝑅𝑅) ⊆ 𝑅)
53, 4sstrd 3755 1 (𝜑 → (𝑆𝑇) ⊆ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3716  ccom 5271
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 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-in 3723  df-ss 3730  df-br 4806  df-opab 4866  df-co 5276
This theorem is referenced by:  trclfvlb2  13971  trrelind  38478  iunrelexpmin1  38521  iunrelexpmin2  38525
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