Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nosgnn0i Structured version   Unicode version

Theorem nosgnn0i 29659
Description: If  X is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.)
Hypothesis
Ref Expression
nosgnn0i.1  |-  X  e. 
{ 1o ,  2o }
Assertion
Ref Expression
nosgnn0i  |-  (/)  =/=  X

Proof of Theorem nosgnn0i
StepHypRef Expression
1 nosgnn0 29658 . . 3  |-  -.  (/)  e.  { 1o ,  2o }
2 nosgnn0i.1 . . . 4  |-  X  e. 
{ 1o ,  2o }
3 eleq1 2526 . . . 4  |-  ( (/)  =  X  ->  ( (/)  e.  { 1o ,  2o } 
<->  X  e.  { 1o ,  2o } ) )
42, 3mpbiri 233 . . 3  |-  ( (/)  =  X  ->  (/)  e.  { 1o ,  2o } )
51, 4mto 176 . 2  |-  -.  (/)  =  X
65neir 2654 1  |-  (/)  =/=  X
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    e. wcel 1823    =/= wne 2649   (/)c0 3783   {cpr 4018   1oc1o 7115   2oc2o 7116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-nul 3784  df-sn 4017  df-pr 4019  df-suc 4873  df-1o 7122  df-2o 7123
This theorem is referenced by:  sltres  29664  nobndlem2  29693  nobndlem4  29695  nobndlem5  29696  nobndlem6  29697  nobndlem8  29699
  Copyright terms: Public domain W3C validator