Theorem unss1 3316

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 3161	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	orim1d 788	. . 3 |- ( A C_ B -> ( ( x e. A \/ x e. C ) -> ( x e. B \/ x e. C ) ) )
3		elun 3288	. . 3 |- ( x e. ( A u. C ) <-> ( x e. A \/ x e. C ) )
4		elun 3288	. . 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 3173	1 |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) )

Syntax hints:   -> wi 4   \/ wo 709   e. wcel 2158   u. cun 3139   C_ wss 3141

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154

This theorem is referenced by:  unss2  3318  unss12  3319  undif1ss  3509  eldifpw  4489  tposss  6260  dftpos4  6277