Theorem ssun1 3136

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 666	. . 3 |- ( x e. A -> ( x e. A \/ x e. B ) )
2		elun 3114	. . 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 3004	1 |- A C_ ( A u. B )

Syntax hints:   \/ wo 662   e. wcel 1434   u. cun 2972   C_ wss 2974

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987

This theorem is referenced by:  ssun2  3137  ssun3  3138  elun1  3140  inabs  3204  reuun1  3253  un00  3297  undifabs  3327  undifss  3330  snsspr1  3541  snsstp1  3543  snsstp2  3544  prsstp12  3546  sssucid  4178  unexb  4203  dmexg  4624  fvun1  5271  dftpos2  5910  tpostpos2  5914  ac6sfi  6431  ressxr  7224  nnssnn0  8358  un0addcl  8388  un0mulcl  8389  nn0ssxnn0  8421  bdunexb  10896