Theorem ssbr 5155

Description: Implication from a subclass relationship of binary relations. (Contributed by Peter Mazsa, 11-Nov-2019.)

Assertion

Ref	Expression
ssbr	⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))

Proof of Theorem ssbr

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
2	1	ssbrd 5154	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))

Syntax hints:   → wi 4   ⊆ wss 3905   class class class wbr 5109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-clel 2838  df-ss 3922  df-br 5110

This theorem is referenced by:  ssbri  5156  coss1  5841  coss2  5842  cnvss  5858  ssrelrn  5884  ttrclss  9685  chnrss  18666  isucn2  24435  brelg  32952  cossss  39184