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Theorem suceq 4104
Description: Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
suceq (A = B → suc A = suc B)

Proof of Theorem suceq
StepHypRef Expression
1 id 19 . . 3 (A = BA = B)
2 sneq 3377 . . 3 (A = B → {A} = {B})
31, 2uneq12d 3092 . 2 (A = B → (A ∪ {A}) = (B ∪ {B}))
4 df-suc 4073 . 2 suc A = (A ∪ {A})
5 df-suc 4073 . 2 suc B = (B ∪ {B})
63, 4, 53eqtr4g 2094 1 (A = B → suc A = suc B)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  cun 2909  {csn 3366  suc csuc 4067
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-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-v 2553  df-un 2916  df-sn 3372  df-suc 4073
This theorem is referenced by:  eqelsuc  4121  onsucsssucexmid  4211  onsucelsucexmidlem  4213  onsucelsucexmid  4214  ordsucunielexmid  4215  suc11g  4234  ordpwsucexmid  4245  peano2  4260  findes  4268  nn0suc  4269  0elnn  4282  frecsuc  5924  sucinc  5957  sucinc2  5958  oacl  5972  oav2  5975  oasuc  5976  oa1suc  5979  nna0r  5989  nnacom  5995  nnaass  5996  nnmsucr  5999  nnsucelsuc  6002  nnsucsssuc  6003  nnaword  6013  nnaordex  6029  indpi  6319  bj-indsuc  8982  bj-bdfindes  9002  bj-nn0suc0  9003  bj-peano4  9008  bj-inf2vnlem1  9019  bj-nn0sucALT  9027  bj-findes  9030
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