Theorem unss1 3240

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 3086	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	orim1d 776	. . 3 |- ( A C_ B -> ( ( x e. A \/ x e. C ) -> ( x e. B \/ x e. C ) ) )
3		elun 3212	. . 3 |- ( x e. ( A u. C ) <-> ( x e. A \/ x e. C ) )
4		elun 3212	. . 3 |- ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
5	2, 3, 4	3imtr4g 204	. 2 |- ( A C_ B -> ( x e. ( A u. C ) -> x e. ( B u. C ) ) )
6	5	ssrdv 3098	1 |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) )

Syntax hints:   -> wi 4   \/ wo 697   e. wcel 1480   u. cun 3064   C_ wss 3066

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079

This theorem is referenced by:  unss2  3242  unss12  3243  undif1ss  3432  eldifpw  4393  tposss  6136  dftpos4  6153