Theorem disjdif 3355

Description: A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
disjdif	|- ( A i^i ( B \ A ) ) = (/)

Proof of Theorem disjdif

Step	Hyp	Ref	Expression
1		inss1 3220	. 2 |- ( A i^i B ) C_ A
2		inssdif0im 3350	. 2 |- ( ( A i^i B ) C_ A -> ( A i^i ( B \ A ) ) = (/) )
3	1, 2	ax-mp 7	1 |- ( A i^i ( B \ A ) ) = (/)

Syntax hints:   = wceq 1289   \ cdif 2996   i^i cin 2998   C_ wss 2999   (/)c0 3286

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287

This theorem is referenced by:  ssdifin0  3364  difdifdirss  3367  fvsnun1  5494  fvsnun2  5495  phplem2  6567  unfiin  6634  xpfi  6638  sbthlem7  6670  sbthlemi8  6671  exmidfodomrlemim  6825  fihashssdif  10222  zfz1isolem1  10241  fsumlessfi  10850  setsfun  11524  setsfun0  11525  setsidn  11539