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Theorem oa1suc 6526
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 6475 . . . 4  |-  1o  =  suc  (/)
21oveq2i 5831 . . 3  |-  ( A  +o  1o )  =  ( A  +o  suc  (/) )
3 peano1 4675 . . . 4  |-  (/)  e.  om
4 onasuc 6523 . . . 4  |-  ( ( A  e.  On  /\  (/) 
e.  om )  ->  ( A  +o  suc  (/) )  =  suc  ( A  +o  (/) ) )
53, 4mpan2 654 . . 3  |-  ( A  e.  On  ->  ( A  +o  suc  (/) )  =  suc  ( A  +o  (/) ) )
62, 5syl5eq 2329 . 2  |-  ( A  e.  On  ->  ( A  +o  1o )  =  suc  ( A  +o  (/) ) )
7 oa0 6511 . . 3  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
8 suceq 4457 . . 3  |-  ( ( A  +o  (/) )  =  A  ->  suc  ( A  +o  (/) )  =  suc  A )
97, 8syl 17 . 2  |-  ( A  e.  On  ->  suc  ( A  +o  (/) )  =  suc  A )
106, 9eqtrd 2317 1  |-  ( A  e.  On  ->  ( A  +o  1o )  =  suc  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   (/)c0 3457   Oncon0 4392   suc csuc 4394   omcom 4656  (class class class)co 5820   1oc1o 6468    +o coa 6472
This theorem is referenced by:  o1p1e2  6535  om1r  6537  omlimcl  6572  oneo  6575  oeeui  6596  nnneo  6645  nneob  6646  oancom  7348  indpi  8527
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479
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