Theorem ssun1 3131

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 641	. . 3 |- ( x e. A -> ( x e. A \/ x e. B ) )
2		elun 3109	. . 3 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
3	1, 2	sylibr 141	. 2 |- ( x e. A -> x e. ( A u. B ) )
4	3	ssriv 2974	1 |- A C_ ( A u. B )

Syntax hints:   \/ wo 637   e. wcel 1407   u. cun 2940   C_ wss 2942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036

This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-un 2947  df-in 2949  df-ss 2956

This theorem is referenced by:  ssun2  3132  ssun3  3133  elun1  3135  inabs  3193  reuun1  3244  un00  3288  undifabs  3325  undifss  3328  snsspr1  3537  snsstp1  3539  snsstp2  3540  prsstp12  3542  sssucid  4177  unexb  4202  dmexg  4621  fvun1  5264  dftpos2  5904  tpostpos2  5908  ac6sfi  6380  ressxr  7098  nnssnn0  8212  un0addcl  8242  un0mulcl  8243  bdunexb  10370