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Theorem suceq 4139
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 3386 . . 3 (𝐴 = 𝐵 → {𝐴} = {𝐵})
31, 2uneq12d 3098 . 2 (𝐴 = 𝐵 → (𝐴 ∪ {𝐴}) = (𝐵 ∪ {𝐵}))
4 df-suc 4108 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
5 df-suc 4108 . 2 suc 𝐵 = (𝐵 ∪ {𝐵})
63, 4, 53eqtr4g 2097 1 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  cun 2915  {csn 3375  suc csuc 4102
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-suc 4108
This theorem is referenced by:  eqelsuc  4156  2ordpr  4249  onsucsssucexmid  4252  onsucelsucexmid  4255  ordsucunielexmid  4256  suc11g  4281  onsucuni2  4288  0elsucexmid  4289  ordpwsucexmid  4294  peano2  4318  findes  4326  nn0suc  4327  0elnn  4340  frecsuc  5991  sucinc  6025  sucinc2  6026  oacl  6040  oav2  6043  oasuc  6044  oa1suc  6047  nna0r  6057  nnacom  6063  nnaass  6064  nnmsucr  6067  nnsucelsuc  6070  nnsucsssuc  6071  nnaword  6084  nnaordex  6100  phplem3g  6319  nneneq  6320  php5  6321  php5dom  6325  indpi  6438  bj-indsuc  10026  bj-bdfindes  10048  bj-nn0suc0  10049  bj-peano4  10054  bj-inf2vnlem1  10069  bj-nn0sucALT  10077  bj-findes  10080
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