Theorem disjdif 3599

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		ssid 3268	. 2 |- A C_ A
2		disjdifg 3598	. 2 |- ( A C_ A -> ( A i^i ( B \ A ) ) = (/) )
3	1, 2	ax-mp 5	1 |- ( A i^i ( B \ A ) ) = (/)

This proof depends on syntax axioms:   = wceq 1402   \ cdif 3217   i^i cin 3219   C_ wss 3220   (/)c0 3520

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521

This theorem is used by:  disjdifr  3600  ssdifin0  3609  difdifdirss  3612  fvsnun1  5912  fvsnun2  5913  phplem2  7154  unfiin  7233  xpfi  7239  sbthlem7  7280  sbthlemi8  7281  exmidfodomrlemim  7553  fihashssdif  11259  hashf1lem2  11286  zfz1isolem1  11292  fsumlessfi  12227  fprodsplit1f  12401  setsfun  13387  setsfun0  13388  setsslid  13403