Theorem ssbri 3379

Description: Inference from a subclass relationship of binary relations. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

Hypothesis

Ref	Expression
ssbri.1	|- A C_ B

Assertion

Ref	Expression
ssbri	|- (CAD -> CBD)

Proof of Theorem ssbri

Step	Hyp	Ref	Expression
1		ssid 2634	. 2 |- A C_ A
2		ssbri.1	. . . 4 |- A C_ B
3	2	a1i 8	. . 3 |- (A C_ A -> A C_ B)
4	3	ssbrd 3378	. 2 |- (A C_ A -> (CAD -> CBD))
5	1, 4	ax-mp 7	1 |- (CAD -> CBD)

Syntax hints:   -> wi 3   C_ wss 2593   class class class wbr 3338

This theorem is referenced by:  endom 5444  brdom3 5963  brdom5 5964  brdom4 5965  fundmpss 13836  fnessref 15503  brresi 15699

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-in 2603  df-ss 2605  df-br 3339

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