Theorem clsslem 14957

Description: The closure of a subclass is a subclass of the closure. (Contributed by RP, 16-May-2020.)

Assertion

Ref	Expression
clsslem	⊢ (𝑅 ⊆ 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} ⊆ ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)})

Distinct variable groups:   𝑅,𝑟   𝑆,𝑟

Allowed substitution hint:   𝜑(𝑟)

Proof of Theorem clsslem

Step	Hyp	Ref	Expression
1		sstr2 3985	. . . 4 ⊢ (𝑅 ⊆ 𝑆 → (𝑆 ⊆ 𝑟 → 𝑅 ⊆ 𝑟))
2	1	anim1d 610	. . 3 ⊢ (𝑅 ⊆ 𝑆 → ((𝑆 ⊆ 𝑟 ∧ 𝜑) → (𝑅 ⊆ 𝑟 ∧ 𝜑)))
3	2	ss2abdv 4056	. 2 ⊢ (𝑅 ⊆ 𝑆 → {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)} ⊆ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)})
4		intss 4967	. 2 ⊢ ({𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)} ⊆ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} ⊆ ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)})
5	3, 4	syl 17	1 ⊢ (𝑅 ⊆ 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} ⊆ ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)})

Syntax hints:   → wi 4   ∧ wa 395  {cab 2705   ⊆ wss 3945  ∩ cint 4944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-v 3472  df-in 3952  df-ss 3962  df-int 4945

This theorem is referenced by:  trclsslem  14963