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Theorem nsuceq0g 4433
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 3441 . . 3  |-  -.  A  e.  (/)
2 sucidg 4431 . . . 4  |-  ( A  e.  V  ->  A  e.  suc  A )
3 eleq2 2253 . . . 4  |-  ( suc 
A  =  (/)  ->  ( A  e.  suc  A  <->  A  e.  (/) ) )
42, 3syl5ibcom 155 . . 3  |-  ( A  e.  V  ->  ( suc  A  =  (/)  ->  A  e.  (/) ) )
51, 4mtoi 665 . 2  |-  ( A  e.  V  ->  -.  suc  A  =  (/) )
65neneqad 2439 1  |-  ( A  e.  V  ->  suc  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160    =/= wne 2360   (/)c0 3437   suc csuc 4380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-v 2754  df-dif 3146  df-un 3148  df-nul 3438  df-sn 3613  df-suc 4386
This theorem is referenced by:  onsucelsucexmid  4544  peano3  4610  frec0g  6416  2on0  6445  zfz1iso  10840
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