Theorem ssun 3351

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 3337	. 2 |- ( A C_ B -> A C_ ( B u. C ) )
2		ssun4 3338	. 2 |- ( A C_ C -> A C_ ( B u. C ) )
3	1, 2	jaoi 717	1 |- ( ( A C_ B \/ A C_ C ) -> A C_ ( B u. C ) )

Syntax hints:   -> wi 4   \/ wo 709   u. cun 3163   C_ wss 3165

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-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178

This theorem is referenced by:  pwunss  4329  pwssunim  4330