Theorem ssbri 5194

Description: Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)

Hypothesis

Ref	Expression
ssbri.1	⊢ 𝐴 ⊆ 𝐵

Assertion

Ref	Expression
ssbri	⊢ (𝐶𝐴𝐷 → 𝐶𝐵𝐷)

Proof of Theorem ssbri

Step	Hyp	Ref	Expression
1		ssbri.1	. 2 ⊢ 𝐴 ⊆ 𝐵
2		ssbr 5193	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))
3	1, 2	ax-mp 5	1 ⊢ (𝐶𝐴𝐷 → 𝐶𝐵𝐷)

Syntax hints:   → wi 4   ⊆ wss 3949   class class class wbr 5149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-br 5150

This theorem is referenced by:  brel  5742  swoer  8733  swoord1  8734  swoord2  8735  ecopover  8815  endom  8975  brdom3  10523  brdom5  10524  brdom4  10525  fpwwe2lem12  10637  nqerf  10925  nqerrel  10927  isfull  17861  isfth  17865  fulloppc  17873  fthoppc  17874  fthsect  17876  fthinv  17877  fthmon  17878  fthepi  17879  ffthiso  17880  catcisolem  18060  psss  18533  efgrelex  19619  hlimadd  30446  hhsscms  30531  occllem  30556  nlelchi  31314  hmopidmchi  31404  fundmpss  34738  itg2gt0cn  36543  brresi2  36588  fullthinc2  47667  thincciso  47669