Theorem disjdif 3567

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 3427	. 2 |- ( A i^i B ) C_ A
2		inssdif0im 3562	. 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 1397   \ cdif 3197   i^i cin 3199   C_ wss 3200   (/)c0 3494

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495

This theorem is referenced by:  ssdifin0  3576  difdifdirss  3579  fvsnun1  5850  fvsnun2  5851  phplem2  7038  unfiin  7117  xpfi  7123  sbthlem7  7161  sbthlemi8  7162  exmidfodomrlemim  7411  fihashssdif  11081  zfz1isolem1  11103  fsumlessfi  12020  fprodsplit1f  12194  setsfun  13116  setsfun0  13117  setsslid  13132