MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nsuceq0 Structured version   Unicode version

Theorem nsuceq0 4664
Description: No successor is empty. (Contributed by NM, 3-Apr-1995.)
Assertion
Ref Expression
nsuceq0  |-  suc  A  =/=  (/)

Proof of Theorem nsuceq0
StepHypRef Expression
1 noel 3634 . . . 4  |-  -.  A  e.  (/)
2 sucidg 4662 . . . . 5  |-  ( A  e.  _V  ->  A  e.  suc  A )
3 eleq2 2499 . . . . 5  |-  ( suc 
A  =  (/)  ->  ( A  e.  suc  A  <->  A  e.  (/) ) )
42, 3syl5ibcom 213 . . . 4  |-  ( A  e.  _V  ->  ( suc  A  =  (/)  ->  A  e.  (/) ) )
51, 4mtoi 172 . . 3  |-  ( A  e.  _V  ->  -.  suc  A  =  (/) )
6 sucprc 4659 . . . . . . 7  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
76eqeq1d 2446 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( suc  A  =  (/)  <->  A  =  (/) ) )
8 0ex 4342 . . . . . . 7  |-  (/)  e.  _V
9 eleq1 2498 . . . . . . 7  |-  ( A  =  (/)  ->  ( A  e.  _V  <->  (/)  e.  _V ) )
108, 9mpbiri 226 . . . . . 6  |-  ( A  =  (/)  ->  A  e. 
_V )
117, 10syl6bi 221 . . . . 5  |-  ( -.  A  e.  _V  ->  ( suc  A  =  (/)  ->  A  e.  _V )
)
1211con3d 128 . . . 4  |-  ( -.  A  e.  _V  ->  ( -.  A  e.  _V  ->  -.  suc  A  =  (/) ) )
1312pm2.43i 46 . . 3  |-  ( -.  A  e.  _V  ->  -. 
suc  A  =  (/) )
145, 13pm2.61i 159 . 2  |-  -.  suc  A  =  (/)
1514neir 2606 1  |-  suc  A  =/=  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958   (/)c0 3630   suc csuc 4586
This theorem is referenced by:  0elsuc  4818  peano3  4869  2on0  6736  oelim2  6841  limenpsi  7285  enp1i  7346  findcard2  7351  fseqdom  7912  dfac12lem2  8029  cfsuc  8142  cfpwsdom  8464  rankcf  8657  dfrdg2  25428  nosgnn0  25618  sltsolem1  25628  dfrdg4  25800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4341
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-nul 3631  df-sn 3822  df-suc 4590
  Copyright terms: Public domain W3C validator