Theorem ssun 3306

Description: A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)

Assertion

Ref	Expression
ssun	|- ( ( A C_ B \/ A C_ C ) -> A C_ ( B u. C ) )

Proof of Theorem ssun

Step	Hyp	Ref	Expression
1		ssun3 3292	. 2 |- ( A C_ B -> A C_ ( B u. C ) )
2		ssun4 3293	. 2 |- ( A C_ C -> A C_ ( B u. C ) )
3	1, 2	jaoi 711	1 |- ( ( A C_ B \/ A C_ C ) -> A C_ ( B u. C ) )

Syntax hints:   -> wi 4   \/ wo 703   u. cun 3119   C_ wss 3121

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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  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-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134

This theorem is referenced by:  pwunss  4268  pwssunim  4269