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Theorem nna0r 6844
 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 6774) so that we can avoid ax-rep 4312, 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

Proof of Theorem nna0r
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6081 . . 3
2 id 20 . . 3
31, 2eqeq12d 2449 . 2
4 oveq2 6081 . . 3
5 id 20 . . 3
64, 5eqeq12d 2449 . 2
7 oveq2 6081 . . 3
8 id 20 . . 3
97, 8eqeq12d 2449 . 2
10 oveq2 6081 . . 3
11 id 20 . . 3
1210, 11eqeq12d 2449 . 2
13 0elon 4626 . . 3
14 oa0 6752 . . 3
1513, 14ax-mp 8 . 2
16 peano1 4856 . . . 4
17 nnasuc 6841 . . . 4
1816, 17mpan 652 . . 3
19 suceq 4638 . . . 4
2019eqeq2d 2446 . . 3
2118, 20syl5ibcom 212 . 2
223, 6, 9, 12, 15, 21finds 4863 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  c0 3620  con0 4573   csuc 4575  com 4837  (class class class)co 6073   coa 6713 This theorem is referenced by:  nnacom  6852  nnm1  6883 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-oadd 6720
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