Theorem disjdif 3520

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 3380	. 2 |- ( A i^i B ) C_ A
2		inssdif0im 3515	. 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 3151   i^i cin 3153   C_ wss 3154   (/)c0 3447

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 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3156  df-in 3160  df-ss 3167  df-nul 3448

This theorem is referenced by:  ssdifin0  3529  difdifdirss  3532  fvsnun1  5756  fvsnun2  5757  phplem2  6911  unfiin  6984  xpfi  6988  sbthlem7  7024  sbthlemi8  7025  exmidfodomrlemim  7263  fihashssdif  10892  zfz1isolem1  10914  fsumlessfi  11606  fprodsplit1f  11780  setsfun  12656  setsfun0  12657  setsslid  12672