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Theorem nna0r 7259
Description: Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. Note: In this and later theorems, we deliberately avoid the more general ordinal versions of these theorems (in this case oa0r 7189) so that we can avoid ax-rep 4558, which is not needed for finite recursive definitions. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
Assertion
Ref Expression
nna0r  |-  ( A  e.  om  ->  ( (/) 
+o  A )  =  A )

Proof of Theorem nna0r
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6293 . . 3  |-  ( x  =  (/)  ->  ( (/)  +o  x )  =  (
(/)  +o  (/) ) )
2 id 22 . . 3  |-  ( x  =  (/)  ->  x  =  (/) )
31, 2eqeq12d 2489 . 2  |-  ( x  =  (/)  ->  ( (
(/)  +o  x )  =  x  <->  ( (/)  +o  (/) )  =  (/) ) )
4 oveq2 6293 . . 3  |-  ( x  =  y  ->  ( (/) 
+o  x )  =  ( (/)  +o  y
) )
5 id 22 . . 3  |-  ( x  =  y  ->  x  =  y )
64, 5eqeq12d 2489 . 2  |-  ( x  =  y  ->  (
( (/)  +o  x )  =  x  <->  ( (/)  +o  y
)  =  y ) )
7 oveq2 6293 . . 3  |-  ( x  =  suc  y  -> 
( (/)  +o  x )  =  ( (/)  +o  suc  y ) )
8 id 22 . . 3  |-  ( x  =  suc  y  ->  x  =  suc  y )
97, 8eqeq12d 2489 . 2  |-  ( x  =  suc  y  -> 
( ( (/)  +o  x
)  =  x  <->  ( (/)  +o  suc  y )  =  suc  y ) )
10 oveq2 6293 . . 3  |-  ( x  =  A  ->  ( (/) 
+o  x )  =  ( (/)  +o  A
) )
11 id 22 . . 3  |-  ( x  =  A  ->  x  =  A )
1210, 11eqeq12d 2489 . 2  |-  ( x  =  A  ->  (
( (/)  +o  x )  =  x  <->  ( (/)  +o  A
)  =  A ) )
13 0elon 4931 . . 3  |-  (/)  e.  On
14 oa0 7167 . . 3  |-  ( (/)  e.  On  ->  ( (/)  +o  (/) )  =  (/) )
1513, 14ax-mp 5 . 2  |-  ( (/)  +o  (/) )  =  (/)
16 peano1 6704 . . . 4  |-  (/)  e.  om
17 nnasuc 7256 . . . 4  |-  ( (
(/)  e.  om  /\  y  e.  om )  ->  ( (/) 
+o  suc  y )  =  suc  ( (/)  +o  y
) )
1816, 17mpan 670 . . 3  |-  ( y  e.  om  ->  ( (/) 
+o  suc  y )  =  suc  ( (/)  +o  y
) )
19 suceq 4943 . . . 4  |-  ( (
(/)  +o  y )  =  y  ->  suc  ( (/) 
+o  y )  =  suc  y )
2019eqeq2d 2481 . . 3  |-  ( (
(/)  +o  y )  =  y  ->  ( (
(/)  +o  suc  y )  =  suc  ( (/)  +o  y )  <->  ( (/)  +o  suc  y )  =  suc  y ) )
2118, 20syl5ibcom 220 . 2  |-  ( y  e.  om  ->  (
( (/)  +o  y )  =  y  ->  ( (/) 
+o  suc  y )  =  suc  y ) )
223, 6, 9, 12, 15, 21finds 6711 1  |-  ( A  e.  om  ->  ( (/) 
+o  A )  =  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 6285   omcom 6685    +o coa 7128
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 6577
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 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-recs 7043  df-rdg 7077  df-oadd 7135
This theorem is referenced by:  nnacom  7267  nnm1  7298
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