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Theorem nna0r 6781
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 6711) so that we can avoid ax-rep 4254, 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 6021 . . 3  |-  ( x  =  (/)  ->  ( (/)  +o  x )  =  (
(/)  +o  (/) ) )
2 id 20 . . 3  |-  ( x  =  (/)  ->  x  =  (/) )
31, 2eqeq12d 2394 . 2  |-  ( x  =  (/)  ->  ( (
(/)  +o  x )  =  x  <->  ( (/)  +o  (/) )  =  (/) ) )
4 oveq2 6021 . . 3  |-  ( x  =  y  ->  ( (/) 
+o  x )  =  ( (/)  +o  y
) )
5 id 20 . . 3  |-  ( x  =  y  ->  x  =  y )
64, 5eqeq12d 2394 . 2  |-  ( x  =  y  ->  (
( (/)  +o  x )  =  x  <->  ( (/)  +o  y
)  =  y ) )
7 oveq2 6021 . . 3  |-  ( x  =  suc  y  -> 
( (/)  +o  x )  =  ( (/)  +o  suc  y ) )
8 id 20 . . 3  |-  ( x  =  suc  y  ->  x  =  suc  y )
97, 8eqeq12d 2394 . 2  |-  ( x  =  suc  y  -> 
( ( (/)  +o  x
)  =  x  <->  ( (/)  +o  suc  y )  =  suc  y ) )
10 oveq2 6021 . . 3  |-  ( x  =  A  ->  ( (/) 
+o  x )  =  ( (/)  +o  A
) )
11 id 20 . . 3  |-  ( x  =  A  ->  x  =  A )
1210, 11eqeq12d 2394 . 2  |-  ( x  =  A  ->  (
( (/)  +o  x )  =  x  <->  ( (/)  +o  A
)  =  A ) )
13 0elon 4568 . . 3  |-  (/)  e.  On
14 oa0 6689 . . 3  |-  ( (/)  e.  On  ->  ( (/)  +o  (/) )  =  (/) )
1513, 14ax-mp 8 . 2  |-  ( (/)  +o  (/) )  =  (/)
16 peano1 4797 . . . 4  |-  (/)  e.  om
17 nnasuc 6778 . . . 4  |-  ( (
(/)  e.  om  /\  y  e.  om )  ->  ( (/) 
+o  suc  y )  =  suc  ( (/)  +o  y
) )
1816, 17mpan 652 . . 3  |-  ( y  e.  om  ->  ( (/) 
+o  suc  y )  =  suc  ( (/)  +o  y
) )
19 suceq 4580 . . . 4  |-  ( (
(/)  +o  y )  =  y  ->  suc  ( (/) 
+o  y )  =  suc  y )
2019eqeq2d 2391 . . 3  |-  ( (
(/)  +o  y )  =  y  ->  ( (
(/)  +o  suc  y )  =  suc  ( (/)  +o  y )  <->  ( (/)  +o  suc  y )  =  suc  y ) )
2118, 20syl5ibcom 212 . 2  |-  ( y  e.  om  ->  (
( (/)  +o  y )  =  y  ->  ( (/) 
+o  suc  y )  =  suc  y ) )
223, 6, 9, 12, 15, 21finds 4804 1  |-  ( A  e.  om  ->  ( (/) 
+o  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   (/)c0 3564   Oncon0 4515   suc csuc 4517   omcom 4778  (class class class)co 6013    +o coa 6650
This theorem is referenced by:  nnacom  6789  nnm1  6820
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-recs 6562  df-rdg 6597  df-oadd 6657
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