Theorem ssun1 3135

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 665	. . 3 |- ( x e. A -> ( x e. A \/ x e. B ) )
2		elun 3113	. . 3 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
3	1, 2	sylibr 132	. 2 |- ( x e. A -> x e. ( A u. B ) )
4	3	ssriv 3003	1 |- A C_ ( A u. B )

Syntax hints:   \/ wo 661   e. wcel 1433   u. cun 2971   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986

This theorem is referenced by:  ssun2  3136  ssun3  3137  elun1  3139  inabs  3197  reuun1  3246  un00  3290  undifabs  3320  undifss  3323  snsspr1  3533  snsstp1  3535  snsstp2  3536  prsstp12  3538  sssucid  4170  unexb  4195  dmexg  4614  fvun1  5260  dftpos2  5899  tpostpos2  5903  ac6sfi  6379  ressxr  7162  nnssnn0  8291  un0addcl  8321  un0mulcl  8322  bdunexb  10711