Theorem clsslem 14950

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 3953	. . . 4 ⊢ (𝑅 ⊆ 𝑆 → (𝑆 ⊆ 𝑟 → 𝑅 ⊆ 𝑟))
2	1	anim1d 611	. . 3 ⊢ (𝑅 ⊆ 𝑆 → ((𝑆 ⊆ 𝑟 ∧ 𝜑) → (𝑅 ⊆ 𝑟 ∧ 𝜑)))
3	2	ss2abdv 4029	. 2 ⊢ (𝑅 ⊆ 𝑆 → {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)} ⊆ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)})
4		intss 4933	. 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 2707   ⊆ wss 3914  ∩ cint 4910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-ral 3045  df-ss 3931  df-int 4911

This theorem is referenced by:  trclsslem  14956