Theorem ssun1 3367

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 717	. . 3 |- ( x e. A -> ( x e. A \/ x e. B ) )
2		elun 3345	. . 3 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
3	1, 2	sylibr 134	. 2 |- ( x e. A -> x e. ( A u. B ) )
4	3	ssriv 3228	1 |- A C_ ( A u. B )

Syntax hints:   \/ wo 713   e. wcel 2200   u. cun 3195   C_ wss 3197

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210

This theorem is referenced by:  ssun2  3368  ssun3  3369  elun1  3371  inabs  3436  reuun1  3486  un00  3538  undifabs  3568  undifss  3572  snsspr1  3816  snsstp1  3818  snsstp2  3819  prsstp12  3821  exmidundif  4290  sssucid  4506  unexb  4533  dmexg  4988  fvun1  5700  dftpos2  6407  tpostpos2  6411  ac6sfi  7060  caserel  7254  finomni  7307  ressxr  8190  nnssnn0  9372  un0addcl  9402  un0mulcl  9403  nn0ssxnn0  9435  ccatclab  11129  ccatrn  11144  fsumsplit  11918  fsum2d  11946  fsumabs  11976  fprodsplitdc  12107  fprod2d  12134  ennnfonelemss  12981  prdssca  13308  prdsbas  13309  prdsplusg  13310  prdsmulr  13311  lspun  14366  cnfldbas  14524  mpocnfldadd  14525  mpocnfldmul  14527  cnfldcj  14529  cnfldtset  14530  cnfldle  14531  cnfldds  14532  psrplusgg  14642  dvmptfsum  15399  elplyr  15414  lgsdir2lem3  15709  lgsquadlem2  15757  bdunexb  16283