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Theorem trrelssd 13758
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 13757 . 2 (𝜑 → (𝑆𝑇) ⊆ (𝑅𝑅))
4 trrelssd.r . 2 (𝜑 → (𝑅𝑅) ⊆ 𝑅)
53, 4sstrd 3646 1 (𝜑 → (𝑆𝑇) ⊆ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3607  ccom 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-in 3614  df-ss 3621  df-br 4686  df-opab 4746  df-co 5152
This theorem is referenced by:  trclfvlb2  13795  trrelind  38274  iunrelexpmin1  38317  iunrelexpmin2  38321
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