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Theorem suceq 4167
 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 3414 . . 3 (𝐴 = 𝐵 → {𝐴} = {𝐵})
31, 2uneq12d 3126 . 2 (𝐴 = 𝐵 → (𝐴 ∪ {𝐴}) = (𝐵 ∪ {𝐵}))
4 df-suc 4136 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
5 df-suc 4136 . 2 suc 𝐵 = (𝐵 ∪ {𝐵})
63, 4, 53eqtr4g 2113 1 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259   ∪ cun 2943  {csn 3403  suc csuc 4130 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-suc 4136 This theorem is referenced by:  eqelsuc  4184  2ordpr  4277  onsucsssucexmid  4280  onsucelsucexmid  4283  ordsucunielexmid  4284  suc11g  4309  onsucuni2  4316  0elsucexmid  4317  ordpwsucexmid  4322  peano2  4346  findes  4354  nn0suc  4355  0elnn  4368  frecsuc  6022  sucinc  6056  sucinc2  6057  oacl  6071  oav2  6074  oasuc  6075  oa1suc  6078  nna0r  6088  nnacom  6094  nnaass  6095  nnmsucr  6098  nnsucelsuc  6101  nnsucsssuc  6102  nnaword  6115  nnaordex  6131  phplem3g  6350  nneneq  6351  php5  6352  php5dom  6356  indpi  6498  bj-indsuc  10439  bj-bdfindes  10461  bj-nn0suc0  10462  bj-peano4  10467  bj-inf2vnlem1  10482  bj-nn0sucALT  10490  bj-findes  10493
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