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Theorem nsuceq0g 4121
Description: No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)
Assertion
Ref Expression
nsuceq0g  V  suc  =/=  (/)

Proof of Theorem nsuceq0g
StepHypRef Expression
1 noel 3222 . . 3  (/)
2 sucidg 4119 . . . 4  V  suc
3 eleq2 2098 . . . 4  suc  (/)  suc  (/)
42, 3syl5ibcom 144 . . 3  V  suc  (/)  (/)
51, 4mtoi 589 . 2  V  suc  (/)
65neneqad 2278 1  V  suc  =/=  (/)
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242   wcel 1390    =/= wne 2201   (/)c0 3218   suc csuc 4068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-un 2916  df-nul 3219  df-sn 3373  df-suc 4074
This theorem is referenced by:  onsucelsucexmid  4215  peano3  4262  frec0g  5922  2on0  5949
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