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Theorem peano5nni 7698
Description: Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
peano5nni  1  +  1 
NN  C_
Distinct variable group:   ,

Proof of Theorem peano5nni
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 1re 6824 . . . 4  1  RR
2 elin 3120 . . . . 5  1  i^i  RR  1  1  RR
32biimpri 124 . . . 4  1  1  RR  1  i^i  RR
41, 3mpan2 401 . . 3  1  1  i^i  RR
5 inss1 3151 . . . . 5  i^i  RR  C_
6 ssralv 2998 . . . . 5  i^i  RR 
C_  +  1  i^i  RR  +  1
75, 6ax-mp 7 . . . 4  +  1  i^i  RR  +  1
8 inss2 3152 . . . . . . . 8  i^i  RR  C_  RR
98sseli 2935 . . . . . . 7  i^i  RR  RR
10 1red 6840 . . . . . . 7  i^i  RR  1  RR
119, 10readdcld 6852 . . . . . 6  i^i  RR  +  1  RR
12 elin 3120 . . . . . . 7  +  1  i^i  RR  +  1  +  1  RR
1312simplbi2com 1330 . . . . . 6  +  1  RR  +  1  +  1  i^i  RR
1411, 13syl 14 . . . . 5  i^i  RR  +  1  +  1  i^i  RR
1514ralimia 2376 . . . 4  i^i  RR  +  1  i^i  RR  +  1  i^i  RR
167, 15syl 14 . . 3  +  1  i^i  RR  +  1  i^i  RR
17 reex 6813 . . . . 5  RR  _V
1817inex2 3883 . . . 4  i^i  RR 
_V
19 eleq2 2098 . . . . . . 7  i^i  RR  1  1  i^i  RR
20 eleq2 2098 . . . . . . . 8  i^i  RR  +  1  +  1  i^i  RR
2120raleqbi1dv 2507 . . . . . . 7  i^i  RR  +  1  i^i  RR  +  1  i^i  RR
2219, 21anbi12d 442 . . . . . 6  i^i  RR  1  +  1  1  i^i  RR  i^i  RR  +  1  i^i  RR
2322elabg 2682 . . . . 5  i^i  RR  _V  i^i  RR  {  |  1  +  1  }  1  i^i  RR  i^i  RR  +  1  i^i  RR
24 dfnn2 7697 . . . . . 6  NN  |^|
{  |  1  +  1  }
25 intss1 3621 . . . . . 6  i^i  RR  {  |  1  +  1  }  |^| {  |  1  +  1  }  C_  i^i  RR
2624, 25syl5eqss 2983 . . . . 5  i^i  RR  {  |  1  +  1  }  NN  C_  i^i  RR
2723, 26syl6bir 153 . . . 4  i^i  RR  _V  1  i^i  RR  i^i  RR  +  1  i^i  RR  NN  C_  i^i  RR
2818, 27ax-mp 7 . . 3  1  i^i  RR  i^i  RR  +  1  i^i  RR  NN  C_  i^i  RR
294, 16, 28syl2an 273 . 2  1  +  1 
NN  C_  i^i  RR
3029, 5syl6ss 2951 1  1  +  1 
NN  C_
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390   {cab 2023  wral 2300   _Vcvv 2551    i^i cin 2910    C_ wss 2911   |^|cint 3606  (class class class)co 5455   RRcr 6710   1c1 6712    + caddc 6714   NNcn 7695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-cnex 6774  ax-resscn 6775  ax-1re 6777  ax-addrcl 6780
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-int 3607  df-inn 7696
This theorem is referenced by:  nnssre  7699  nnind  7711
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