MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0inp0 Structured version   Unicode version

Theorem 0inp0 4609
Description: Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 21-Jun-1993.)
Assertion
Ref Expression
0inp0  |-  ( A  =  (/)  ->  -.  A  =  { (/) } )

Proof of Theorem 0inp0
StepHypRef Expression
1 0nep0 4608 . . 3  |-  (/)  =/=  { (/)
}
2 neeq1 2724 . . 3  |-  ( A  =  (/)  ->  ( A  =/=  { (/) }  <->  (/)  =/=  { (/)
} ) )
31, 2mpbiri 233 . 2  |-  ( A  =  (/)  ->  A  =/= 
{ (/) } )
43neneqd 2645 1  |-  ( A  =  (/)  ->  -.  A  =  { (/) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1383    =/= wne 2638   (/)c0 3770   {csn 4014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-nul 4566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-v 3097  df-dif 3464  df-nul 3771  df-sn 4015
This theorem is referenced by:  dtruALT  4669  zfpair  4674  dtruALT2  4681
  Copyright terms: Public domain W3C validator