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Theorem clsslem 14933
Description: The closure of a subclass is a subclass of the closure. (Contributed by RP, 16-May-2020.)
Assertion
Ref Expression
clsslem (𝑅𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} ⊆ {𝑟 ∣ (𝑆𝑟𝜑)})
Distinct variable groups:   𝑅,𝑟   𝑆,𝑟
Allowed substitution hint:   𝜑(𝑟)

Proof of Theorem clsslem
StepHypRef Expression
1 sstr2 3982 . . . 4 (𝑅𝑆 → (𝑆𝑟𝑅𝑟))
21anim1d 610 . . 3 (𝑅𝑆 → ((𝑆𝑟𝜑) → (𝑅𝑟𝜑)))
32ss2abdv 4053 . 2 (𝑅𝑆 → {𝑟 ∣ (𝑆𝑟𝜑)} ⊆ {𝑟 ∣ (𝑅𝑟𝜑)})
4 intss 4964 . 2 ({𝑟 ∣ (𝑆𝑟𝜑)} ⊆ {𝑟 ∣ (𝑅𝑟𝜑)} → {𝑟 ∣ (𝑅𝑟𝜑)} ⊆ {𝑟 ∣ (𝑆𝑟𝜑)})
53, 4syl 17 1 (𝑅𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} ⊆ {𝑟 ∣ (𝑆𝑟𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  {cab 2701  wss 3941   cint 4941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-v 3468  df-in 3948  df-ss 3958  df-int 4942
This theorem is referenced by:  trclsslem  14939
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