MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difidALT Unicode version

Theorem difidALT 3689
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 3688. (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 3321 . 2  |-  ( A 
\  A )  =  { x  e.  A  |  -.  x  e.  A }
2 dfnul3 3623 . 2  |-  (/)  =  {
x  e.  A  |  -.  x  e.  A }
31, 2eqtr4i 2458 1  |-  ( A 
\  A )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   {crab 2701    \ cdif 3309   (/)c0 3620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-nul 3621
  Copyright terms: Public domain W3C validator