Theorem trclsslem 13939

Description: The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.)

Assertion

Ref	Expression
trclsslem	⊢ (𝑅 ⊆ 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})

Distinct variable groups:   𝑅,𝑟   𝑆,𝑟

Proof of Theorem trclsslem

Step	Hyp	Ref	Expression
1		clsslem 13933	1 ⊢ (𝑅 ⊆ 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})

Syntax hints:   → wi 4   ∧ wa 382  {cab 2757   ⊆ wss 3723  ∩ cint 4611   ∘ ccom 5253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-in 3730  df-ss 3737  df-int 4612

This theorem is referenced by:  trclfvss  13955