Theorem difss 3099

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 3095	. 2 |- ( x e. ( A \ B ) -> x e. A )
2	1	ssriv 3004	1 |- ( A \ B ) C_ A

Syntax hints:   \ cdif 2971   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-in1 577  ax-in2 578  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-dif 2976  df-in 2980  df-ss 2987

This theorem is referenced by:  difssd  3100  difss2  3101  ssdifss  3103  0dif  3316  undif1ss  3319  undifabs  3321  inundifss  3322  undifss  3324  unidif  3635  iunxdif2  3728  difexg  3921  reldif  4479  cnvdif  4754  resdif  5173  fndmdif  5298  swoer  6193  swoord1  6194  swoord2  6195  phplem2  6378  phpm  6390  unfiin  6434  pinn  6550  niex  6553  dmaddpi  6566  dmmulpi  6567  lerelxr  7231