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Theorem nsuceq0g 4181
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 3262 . . 3  |-  -.  A  e.  (/)
2 sucidg 4179 . . . 4  |-  ( A  e.  V  ->  A  e.  suc  A )
3 eleq2 2143 . . . 4  |-  ( suc 
A  =  (/)  ->  ( A  e.  suc  A  <->  A  e.  (/) ) )
42, 3syl5ibcom 153 . . 3  |-  ( A  e.  V  ->  ( suc  A  =  (/)  ->  A  e.  (/) ) )
51, 4mtoi 623 . 2  |-  ( A  e.  V  ->  -.  suc  A  =  (/) )
65neneqad 2325 1  |-  ( A  e.  V  ->  suc  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434    =/= wne 2246   (/)c0 3258   suc csuc 4128
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-v 2604  df-dif 2976  df-un 2978  df-nul 3259  df-sn 3412  df-suc 4134
This theorem is referenced by:  onsucelsucexmid  4281  peano3  4345  frec0g  6046  2on0  6074
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