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Theorem difid 3522
Description: The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.)
Assertion
Ref Expression
difid  |-  ( A 
\  A )  =  (/)

Proof of Theorem difid
StepHypRef Expression
1 ssid 3197 . 2  |-  A  C_  A
2 ssdif0 3513 . 2  |-  ( A 
C_  A  <->  ( A  \  A )  =  (/) )
31, 2mpbi 199 1  |-  ( A 
\  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    \ cdif 3149    C_ wss 3152   (/)c0 3455
This theorem is referenced by:  dif0  3524  difun2  3533  difxp1  6154  difxp2  6155  2oconcl  6502  oev2  6522  fin1a2lem13  8038  ruclem13  12520  strle1  13239  efgi1  15030  frgpuptinv  15080  dprdsn  15271  ablfac1eulem  15307  fctop  16741  cctop  16743  topcld  16772  indiscld  16828  mretopd  16829  restcld  16903  conndisj  17142  csdfil  17589  ufinffr  17624  prdsxmslem2  18075  bcth3  18753  voliunlem3  18909  zrdivrng  21099  ballotlemfp1  23050  symdifid  23781  onint1  24299  compne  27054  difprsneq  27480  diftpsneq  27481  usgra0v  27516  frgra1v  27538  1vwmgra  27543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456
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