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Theorem trclsslem 15026
Description: The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.)
Assertion
Ref Expression
trclsslem (𝑅𝑆 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
Distinct variable groups:   𝑅,𝑟   𝑆,𝑟

Proof of Theorem trclsslem
StepHypRef Expression
1 clsslem 15020 1 (𝑅𝑆 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  {cab 2712  wss 3963   cint 4951  ccom 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-ral 3060  df-ss 3980  df-int 4952
This theorem is referenced by:  trclfvss  15042
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