Theorem ssbri 4128

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 4126	. 2 ⊢ (⊤ → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))
4	3	mptru 1404	1 ⊢ (𝐶𝐴𝐷 → 𝐶𝐵𝐷)

Syntax hints:   → wi 4  ⊤wtru 1396   ⊆ wss 3197   class class class wbr 4083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210  df-br 4084

This theorem is referenced by:  brel  4771  swoer  6716  swoord1  6717  swoord2  6718  ecopover  6788  ecopoverg  6791  endom  6922