Theorem difss 3098

Description: Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)

Assertion

Ref	Expression
difss	|- ( A \ B ) C_ A

Proof of Theorem difss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifi 3094	. 2 |- ( x e. ( A \ B ) -> x e. A )
2	1	ssriv 3003	1 |- ( A \ B ) C_ A

Syntax hints:   \ cdif 2970   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-in1 576  ax-in2 577  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-dif 2975  df-in 2979  df-ss 2986

This theorem is referenced by:  difssd  3099  difss2  3100  ssdifss  3102  0dif  3315  undif1ss  3318  undifabs  3320  inundifss  3321  undifss  3323  unidif  3633  iunxdif2  3726  difexg  3919  reldif  4475  cnvdif  4750  resdif  5168  fndmdif  5293  swoer  6157  swoord1  6158  swoord2  6159  phplem2  6339  phpm  6351  pinn  6499  niex  6502  dmaddpi  6515  dmmulpi  6516  lerelxr  7175