Theorem ssbr 5157

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 5156	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3906   class class class wbr 5111

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 1943  ax-6 2000  ax-7 2041  ax-8 2148

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clel 2840  df-ss 3923  df-br 5112

This theorem is used by:  ssbri  5158  coss1  5843  coss2  5844  cnvss  5860  ssrelrn  5886  ttrclss  9696  chnrss  18695  isucn2  24488  brelg  33025  cossss  39224