Theorem disjdif 3496

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 3356	. 2 |- ( A i^i B ) C_ A
2		inssdif0im 3491	. 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 1353   \ cdif 3127   i^i cin 3129   C_ wss 3130   (/)c0 3423

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-dif 3132  df-in 3136  df-ss 3143  df-nul 3424

This theorem is referenced by:  ssdifin0  3505  difdifdirss  3508  fvsnun1  5714  fvsnun2  5715  phplem2  6853  unfiin  6925  xpfi  6929  sbthlem7  6962  sbthlemi8  6963  exmidfodomrlemim  7200  fihashssdif  10798  zfz1isolem1  10820  fsumlessfi  11468  fprodsplit1f  11642  setsfun  12497  setsfun0  12498  setsslid  12513