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

Theorem difidALT 3523
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 3522. (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 3161 . 2  |-  ( A 
\  A )  =  { x  e.  A  |  -.  x  e.  A }
2 dfnul3 3458 . 2  |-  (/)  =  {
x  e.  A  |  -.  x  e.  A }
31, 2eqtr4i 2306 1  |-  ( A 
\  A )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   {crab 2547    \ cdif 3149   (/)c0 3455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-nul 3456
  Copyright terms: Public domain W3C validator