MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oa1suc Structured version   Unicode version

Theorem oa1suc 7181
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 7130 . . . 4  |-  1o  =  suc  (/)
21oveq2i 6295 . . 3  |-  ( A  +o  1o )  =  ( A  +o  suc  (/) )
3 peano1 6703 . . . 4  |-  (/)  e.  om
4 onasuc 7178 . . . 4  |-  ( ( A  e.  On  /\  (/) 
e.  om )  ->  ( A  +o  suc  (/) )  =  suc  ( A  +o  (/) ) )
53, 4mpan2 671 . . 3  |-  ( A  e.  On  ->  ( A  +o  suc  (/) )  =  suc  ( A  +o  (/) ) )
62, 5syl5eq 2520 . 2  |-  ( A  e.  On  ->  ( A  +o  1o )  =  suc  ( A  +o  (/) ) )
7 oa0 7166 . . 3  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
8 suceq 4943 . . 3  |-  ( ( A  +o  (/) )  =  A  ->  suc  ( A  +o  (/) )  =  suc  A )
97, 8syl 16 . 2  |-  ( A  e.  On  ->  suc  ( A  +o  (/) )  =  suc  A )
106, 9eqtrd 2508 1  |-  ( A  e.  On  ->  ( A  +o  1o )  =  suc  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   (/)c0 3785   Oncon0 4878   suc csuc 4880  (class class class)co 6284   omcom 6684   1oc1o 7123    +o coa 7127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134
This theorem is referenced by:  o1p1e2  7190  om1r  7192  omlimcl  7227  oneo  7230  oeeui  7251  nnneo  7300  nneob  7301  oancom  8068  indpi  9285
  Copyright terms: Public domain W3C validator