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Theorem difidALT 3400
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 3399. (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 3047 . 2  |-  ( A 
\  A )  =  { x  e.  A  |  -.  x  e.  A }
2 dfnul3 3334 . 2  |-  (/)  =  {
x  e.  A  |  -.  x  e.  A }
31, 2eqtr4i 2139 1  |-  ( A 
\  A )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1314    e. wcel 1463   {crab 2395    \ cdif 3036   (/)c0 3331
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400  df-v 2660  df-dif 3041  df-nul 3332
This theorem is referenced by: (None)
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