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Theorem nsuceq0g 4378
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 3398 . . 3  |-  -.  A  e.  (/)
2 sucidg 4376 . . . 4  |-  ( A  e.  V  ->  A  e.  suc  A )
3 eleq2 2221 . . . 4  |-  ( suc 
A  =  (/)  ->  ( A  e.  suc  A  <->  A  e.  (/) ) )
42, 3syl5ibcom 154 . . 3  |-  ( A  e.  V  ->  ( suc  A  =  (/)  ->  A  e.  (/) ) )
51, 4mtoi 654 . 2  |-  ( A  e.  V  ->  -.  suc  A  =  (/) )
65neneqad 2406 1  |-  ( A  e.  V  ->  suc  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335    e. wcel 2128    =/= wne 2327   (/)c0 3394   suc csuc 4325
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-v 2714  df-dif 3104  df-un 3106  df-nul 3395  df-sn 3566  df-suc 4331
This theorem is referenced by:  onsucelsucexmid  4489  peano3  4555  frec0g  6344  2on0  6373  zfz1iso  10712
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