Theorem ssbr 5183

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 22	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
2	1	ssbrd 5182	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))

Syntax hints:   → wi 4   ⊆ wss 3941   class class class wbr 5139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-ss 3958  df-br 5140

This theorem is referenced by:  ssbri  5184  coss1  5846  coss2  5847  cnvss  5863  ssrelrn  5885  ttrclss  9712  isucn2  24108  brelg  32310  cossss  37789