ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  difidALT Unicode version

Theorem difidALT 3358
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 3357. (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 3010 . 2  |-  ( A 
\  A )  =  { x  e.  A  |  -.  x  e.  A }
2 dfnul3 3292 . 2  |-  (/)  =  {
x  e.  A  |  -.  x  e.  A }
31, 2eqtr4i 2112 1  |-  ( A 
\  A )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1290    e. wcel 1439   {crab 2364    \ cdif 2999   (/)c0 3289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rab 2369  df-v 2624  df-dif 3004  df-nul 3290
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator