Theorem unss1 3433

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 3268	. . . 4 |- (A C_ B -> (x e. A -> x e. B))
2	1	orim1d 812	. . 3 |- (A C_ B -> ((x e. A \/ x e. C) -> (x e. B \/ x e. C)))
3		elun 3221	. . 3 |- (x e. (A u. C) <-> (x e. A \/ x e. C))
4		elun 3221	. . 3 |- (x e. (B u. C) <-> (x e. B \/ x e. C))
5	2, 3, 4	3imtr4g 261	. 2 |- (A C_ B -> (x e. (A u. C) -> x e. (B u. C)))
6	5	ssrdv 3279	1 |- (A C_ B -> (A u. C) C_ (B u. C))

Syntax hints:   -> wi 4   \/ wo 357   e. wcel 1710   u. cun 3208   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260

This theorem is referenced by:  unss2  3435  unss12  3436  pwadjoin  4120