Theorem clsslem 15180

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 3920	. . . 4 ⊢ (𝑅 ⊆ 𝑆 → (𝑆 ⊆ 𝑟 → 𝑅 ⊆ 𝑟))
2	1	anim1d 623	. . 3 ⊢ (𝑅 ⊆ 𝑆 → ((𝑆 ⊆ 𝑟 ∧ 𝜑) → (𝑅 ⊆ 𝑟 ∧ 𝜑)))
3	2	ss2abdv 3993	. 2 ⊢ (𝑅 ⊆ 𝑆 → {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)} ⊆ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)})
4		intss 4934	. 2 ⊢ ({𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)} ⊆ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} ⊆ ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)})
5	3, 4	syl 18	1 ⊢ (𝑅 ⊆ 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} ⊆ ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  {cab 2741   ⊆ wss 3900  ∩ cint 4912

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-ral 3081  df-ss 3901  df-int 4913

This theorem is used by:  trclsslem  15186