Theorem disjdif 3514

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 3374	. 2 |- ( A i^i B ) C_ A
2		inssdif0im 3509	. 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 1364   \ cdif 3145   i^i cin 3147   C_ wss 3148   (/)c0 3441

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2758  df-dif 3150  df-in 3154  df-ss 3161  df-nul 3442

This theorem is referenced by:  ssdifin0  3523  difdifdirss  3526  fvsnun1  5741  fvsnun2  5742  phplem2  6889  unfiin  6962  xpfi  6966  sbthlem7  7001  sbthlemi8  7002  exmidfodomrlemim  7240  fihashssdif  10845  zfz1isolem1  10867  fsumlessfi  11515  fprodsplit1f  11689  setsfun  12564  setsfun0  12565  setsslid  12580