Theorem disjdif 3532

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 3392	. 2 |- ( A i^i B ) C_ A
2		inssdif0im 3527	. 2 |- ( ( A i^i B ) C_ A -> ( A i^i ( B \ A ) ) = (/) )
3	1, 2	ax-mp 5	1 |- ( A i^i ( B \ A ) ) = (/)

Syntax hints:   = wceq 1372   \ cdif 3162   i^i cin 3164   C_ wss 3165   (/)c0 3459

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-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-in 3171  df-ss 3178  df-nul 3460

This theorem is referenced by:  ssdifin0  3541  difdifdirss  3544  fvsnun1  5780  fvsnun2  5781  phplem2  6949  unfiin  7022  xpfi  7028  sbthlem7  7064  sbthlemi8  7065  exmidfodomrlemim  7308  fihashssdif  10961  zfz1isolem1  10983  fsumlessfi  11742  fprodsplit1f  11916  setsfun  12838  setsfun0  12839  setsslid  12854