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Theorem difidALT 3525
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. (Alternate proof of difid 3524 suggested by David Abernethy, 17-Jun-2012.) (Contributed by NM, 17-Jun-2012.) (Proof modification is discouraged.)
Assertion
Ref Expression
difidALT  |-  ( A 
\  A )  =  (/)
Dummy variable  x is distinct from all other variables.

Proof of Theorem difidALT
StepHypRef Expression
1 dfdif2 3163 . 2  |-  ( A 
\  A )  =  { x  e.  A  |  -.  x  e.  A }
2 dfnul3 3460 . 2  |-  (/)  =  {
x  e.  A  |  -.  x  e.  A }
31, 2eqtr4i 2308 1  |-  ( A 
\  A )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 5    = wceq 1624    e. wcel 1685   {crab 2549    \ cdif 3151   (/)c0 3457
This theorem is referenced by:  fin1a2lem13  8034
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-rab 2554  df-v 2792  df-dif 3157  df-nul 3458
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