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Theorem nsuceq0g 4335
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 3362 . . 3  |-  -.  A  e.  (/)
2 sucidg 4333 . . . 4  |-  ( A  e.  V  ->  A  e.  suc  A )
3 eleq2 2201 . . . 4  |-  ( suc 
A  =  (/)  ->  ( A  e.  suc  A  <->  A  e.  (/) ) )
42, 3syl5ibcom 154 . . 3  |-  ( A  e.  V  ->  ( suc  A  =  (/)  ->  A  e.  (/) ) )
51, 4mtoi 653 . 2  |-  ( A  e.  V  ->  -.  suc  A  =  (/) )
65neneqad 2385 1  |-  ( A  e.  V  ->  suc  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480    =/= wne 2306   (/)c0 3358   suc csuc 4282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-v 2683  df-dif 3068  df-un 3070  df-nul 3359  df-sn 3528  df-suc 4288
This theorem is referenced by:  onsucelsucexmid  4440  peano3  4505  frec0g  6287  2on0  6316  zfz1iso  10577
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