Theorem ssbri 4033

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	. . . 4 ⊢ 𝐴 ⊆ 𝐵
2	1	a1i 9	. . 3 ⊢ (⊤ → 𝐴 ⊆ 𝐵)
3	2	ssbrd 4032	. 2 ⊢ (⊤ → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))
4	3	mptru 1357	1 ⊢ (𝐶𝐴𝐷 → 𝐶𝐵𝐷)

Syntax hints:   → wi 4  ⊤wtru 1349   ⊆ wss 3121   class class class wbr 3989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-br 3990

This theorem is referenced by:  brel  4663  swoer  6541  swoord1  6542  swoord2  6543  ecopover  6611  ecopoverg  6614  endom  6741