Theorem unss1 3304

Description: Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
unss1	|- ( A C_ B -> ( A u. C ) C_ ( B u. C ) )

Proof of Theorem unss1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3149	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	orim1d 787	. . 3 |- ( A C_ B -> ( ( x e. A \/ x e. C ) -> ( x e. B \/ x e. C ) ) )
3		elun 3276	. . 3 |- ( x e. ( A u. C ) <-> ( x e. A \/ x e. C ) )
4		elun 3276	. . 3 |- ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
5	2, 3, 4	3imtr4g 205	. 2 |- ( A C_ B -> ( x e. ( A u. C ) -> x e. ( B u. C ) ) )
6	5	ssrdv 3161	1 |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) )

Syntax hints:   -> wi 4   \/ wo 708   e. wcel 2148   u. cun 3127   C_ wss 3129

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142

This theorem is referenced by:  unss2  3306  unss12  3307  undif1ss  3497  eldifpw  4476  tposss  6244  dftpos4  6261