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Theorem suceq 4284
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  =  B  ->  A  =  B )
2 sneq 3504 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
31, 2uneq12d 3197 . 2  |-  ( A  =  B  ->  ( A  u.  { A } )  =  ( B  u.  { B } ) )
4 df-suc 4253 . 2  |-  suc  A  =  ( A  u.  { A } )
5 df-suc 4253 . 2  |-  suc  B  =  ( B  u.  { B } )
63, 4, 53eqtr4g 2172 1  |-  ( A  =  B  ->  suc  A  =  suc  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    u. cun 3035   {csn 3493   suc csuc 4247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-sn 3499  df-suc 4253
This theorem is referenced by:  eqelsuc  4301  2ordpr  4399  onsucsssucexmid  4402  onsucelsucexmid  4405  ordsucunielexmid  4406  suc11g  4432  onsucuni2  4439  0elsucexmid  4440  ordpwsucexmid  4445  peano2  4469  findes  4477  nn0suc  4478  0elnn  4492  omsinds  4495  tfr1onlemsucaccv  6192  tfrcllemsucaccv  6205  tfrcl  6215  frecabcl  6250  frecsuc  6258  sucinc  6295  sucinc2  6296  oacl  6310  oav2  6313  oasuc  6314  oa1suc  6317  nna0r  6328  nnacom  6334  nnaass  6335  nnmsucr  6338  nnsucelsuc  6341  nnsucsssuc  6342  nnaword  6361  nnaordex  6377  phplem3g  6703  nneneq  6704  php5  6705  php5dom  6710  omp1eomlem  6931  omp1eom  6932  indpi  7098  ennnfoneleminc  11769  ennnfonelemex  11772  bj-indsuc  12818  bj-bdfindes  12839  bj-nn0suc0  12840  bj-peano4  12845  bj-inf2vnlem1  12860  bj-nn0sucALT  12868  bj-findes  12871  nnsf  12891  nninfalllemn  12894  nninfsellemdc  12898  nninfself  12901  nninfsellemeqinf  12904  nninfomni  12907
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