Theorem ssbri 4682

Description: Inference from a subclass relationship of binary relations. (The proof was shortened by Andrew Salmon, 9-Jul-2011.) (Contributed by NM, 28-Mar-2007.) (Revised by set.mm contributors, 9-Jul-2011.)

Hypothesis

Ref	Expression
ssbri.1	⊢ A ⊆ B

Assertion

Ref	Expression
ssbri	⊢ (CAD → CBD)

Proof of Theorem ssbri

Step	Hyp	Ref	Expression
1		ssid 3291	. 2 ⊢ A ⊆ A
2		ssbri.1	. . . 4 ⊢ A ⊆ B
3	2	a1i 10	. . 3 ⊢ (A ⊆ A → A ⊆ B)
4	3	ssbrd 4681	. 2 ⊢ (A ⊆ A → (CAD → CBD))
5	1, 4	ax-mp 5	1 ⊢ (CAD → CBD)

Syntax hints:   → wi 4   ⊆ wss 3258   class class class wbr 4640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-br 4641

This theorem is referenced by:  brel  4831