Theorem disjdif 3466

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 3327	. 2 |- ( A i^i B ) C_ A
2		inssdif0im 3461	. 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 1335   \ cdif 3099   i^i cin 3101   C_ wss 3102   (/)c0 3394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395

This theorem is referenced by:  ssdifin0  3475  difdifdirss  3478  fvsnun1  5661  fvsnun2  5662  phplem2  6791  unfiin  6863  xpfi  6867  sbthlem7  6900  sbthlemi8  6901  exmidfodomrlemim  7119  fihashssdif  10674  zfz1isolem1  10693  fsumlessfi  11339  fprodsplit1f  11513  setsfun  12185  setsfun0  12186  setsslid  12200