Theorem unss1 3250

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 3096	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	orim1d 777	. . 3 |- ( A C_ B -> ( ( x e. A \/ x e. C ) -> ( x e. B \/ x e. C ) ) )
3		elun 3222	. . 3 |- ( x e. ( A u. C ) <-> ( x e. A \/ x e. C ) )
4		elun 3222	. . 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 3108	1 |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) )

Syntax hints:   -> wi 4   \/ wo 698   e. wcel 1481   u. cun 3074   C_ wss 3076

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089

This theorem is referenced by:  unss2  3252  unss12  3253  undif1ss  3442  eldifpw  4406  tposss  6151  dftpos4  6168