Theorem disjdif 3524

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 3384	. 2 |- ( A i^i B ) C_ A
2		inssdif0im 3519	. 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 3154   i^i cin 3156   C_ wss 3157   (/)c0 3451

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3452

This theorem is referenced by:  ssdifin0  3533  difdifdirss  3536  fvsnun1  5762  fvsnun2  5763  phplem2  6923  unfiin  6996  xpfi  7002  sbthlem7  7038  sbthlemi8  7039  exmidfodomrlemim  7280  fihashssdif  10927  zfz1isolem1  10949  fsumlessfi  11642  fprodsplit1f  11816  setsfun  12738  setsfun0  12739  setsslid  12754