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Theorem clsslem 14323
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 3950 . . . 4 (𝑅𝑆 → (𝑆𝑟𝑅𝑟))
21anim1d 613 . . 3 (𝑅𝑆 → ((𝑆𝑟𝜑) → (𝑅𝑟𝜑)))
32ss2abdv 4020 . 2 (𝑅𝑆 → {𝑟 ∣ (𝑆𝑟𝜑)} ⊆ {𝑟 ∣ (𝑅𝑟𝜑)})
4 intss 4870 . 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 399  {cab 2799  wss 3910   cint 4849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-v 3473  df-in 3917  df-ss 3927  df-int 4850
This theorem is referenced by:  trclsslem  14329
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