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Theorem trrelssd 14309
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 14308 . 2 (𝜑 → (𝑆𝑇) ⊆ (𝑅𝑅))
4 trrelssd.r . 2 (𝜑 → (𝑅𝑅) ⊆ 𝑅)
53, 4sstrd 3952 1 (𝜑 → (𝑆𝑇) ⊆ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3909  ccom 5531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-v 3472  df-in 3916  df-ss 3926  df-br 5039  df-opab 5101  df-co 5536
This theorem is referenced by:  trclfvlb2  14346  trrelind  40141  iunrelexpmin1  40184  iunrelexpmin2  40188
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