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Theorem nsuceq0g 4236
Description: No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)
Assertion
Ref Expression
nsuceq0g  |-  ( A  e.  V  ->  suc  A  =/=  (/) )

Proof of Theorem nsuceq0g
StepHypRef Expression
1 noel 3288 . . 3  |-  -.  A  e.  (/)
2 sucidg 4234 . . . 4  |-  ( A  e.  V  ->  A  e.  suc  A )
3 eleq2 2151 . . . 4  |-  ( suc 
A  =  (/)  ->  ( A  e.  suc  A  <->  A  e.  (/) ) )
42, 3syl5ibcom 153 . . 3  |-  ( A  e.  V  ->  ( suc  A  =  (/)  ->  A  e.  (/) ) )
51, 4mtoi 625 . 2  |-  ( A  e.  V  ->  -.  suc  A  =  (/) )
65neneqad 2334 1  |-  ( A  e.  V  ->  suc  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438    =/= wne 2255   (/)c0 3284   suc csuc 4183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-dif 2999  df-un 3001  df-nul 3285  df-sn 3447  df-suc 4189
This theorem is referenced by:  onsucelsucexmid  4336  peano3  4401  frec0g  6144  2on0  6173  zfz1iso  10211
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