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Theorem oa1suc 7117
Description: Addition with 1 is same as successor. Proposition 4.34(a) of [Mendelson] p. 266. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
oa1suc  |-  ( A  e.  On  ->  ( A  +o  1o )  =  suc  A )

Proof of Theorem oa1suc
StepHypRef Expression
1 df-1o 7066 . . . 4  |-  1o  =  suc  (/)
21oveq2i 6225 . . 3  |-  ( A  +o  1o )  =  ( A  +o  suc  (/) )
3 peano1 6636 . . . 4  |-  (/)  e.  om
4 onasuc 7114 . . . 4  |-  ( ( A  e.  On  /\  (/) 
e.  om )  ->  ( A  +o  suc  (/) )  =  suc  ( A  +o  (/) ) )
53, 4mpan2 669 . . 3  |-  ( A  e.  On  ->  ( A  +o  suc  (/) )  =  suc  ( A  +o  (/) ) )
62, 5syl5eq 2445 . 2  |-  ( A  e.  On  ->  ( A  +o  1o )  =  suc  ( A  +o  (/) ) )
7 oa0 7102 . . 3  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
8 suceq 4870 . . 3  |-  ( ( A  +o  (/) )  =  A  ->  suc  ( A  +o  (/) )  =  suc  A )
97, 8syl 16 . 2  |-  ( A  e.  On  ->  suc  ( A  +o  (/) )  =  suc  A )
106, 9eqtrd 2433 1  |-  ( A  e.  On  ->  ( A  +o  1o )  =  suc  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1836   (/)c0 3724   Oncon0 4805   suc csuc 4807  (class class class)co 6214   omcom 6617   1oc1o 7059    +o coa 7063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-reu 2749  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-om 6618  df-recs 6978  df-rdg 7012  df-1o 7066  df-oadd 7070
This theorem is referenced by:  o1p1e2  7126  om1r  7128  omlimcl  7163  oneo  7166  oeeui  7187  nnneo  7236  nneob  7237  oancom  8000  indpi  9214
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