Theorem ssun1 3426

Description: Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ssun1	|- A C_ (A u. B)

Proof of Theorem ssun1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orc 374	. . 3 |- (x e. A -> (x e. A \/ x e. B))
2		elun 3220	. . 3 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
3	1, 2	sylibr 203	. 2 |- (x e. A -> x e. (A u. B))
4	3	ssriv 3277	1 |- A C_ (A u. B)

Syntax hints:   \/ wo 357   e. wcel 1710   u. cun 3207   C_ wss 3257

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259

This theorem is referenced by:  ssun2  3427  ssun3  3428  elun1  3430  inabs  3486  reuun1  3537  un00  3586  snsspr1  3856  snsstp1  3858  snsstp2  3859  uniintsn  3963  snprss1  4120  pw1equn  4331  pw1eqadj  4332  evenoddnnnul  4514  sfinltfin  4535  vfinspsslem1  4550  vfinspss  4551  phi011lem1  4598  enadjlem1  6059  ncdisjun  6136  ce0addcnnul  6179  addlec  6208