Theorem unss1 3376

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 3221	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	orim1d 794	. . 3 |- ( A C_ B -> ( ( x e. A \/ x e. C ) -> ( x e. B \/ x e. C ) ) )
3		elun 3348	. . 3 |- ( x e. ( A u. C ) <-> ( x e. A \/ x e. C ) )
4		elun 3348	. . 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 3233	1 |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) )

Syntax hints:   -> wi 4   \/ wo 715   e. wcel 2202   u. cun 3198   C_ wss 3200

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213

This theorem is referenced by:  unss2  3378  unss12  3379  undif1ss  3569  eldifpw  4574  tposss  6411  dftpos4  6428  plyss  15461