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Theorem difidALT 3402
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 3401. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
difidALT  |-  ( A 
\  A )  =  (/)

Proof of Theorem difidALT
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dfdif2 3049 . 2  |-  ( A 
\  A )  =  { x  e.  A  |  -.  x  e.  A }
2 dfnul3 3336 . 2  |-  (/)  =  {
x  e.  A  |  -.  x  e.  A }
31, 2eqtr4i 2141 1  |-  ( A 
\  A )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1316    e. wcel 1465   {crab 2397    \ cdif 3038   (/)c0 3333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-dif 3043  df-nul 3334
This theorem is referenced by: (None)
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