ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  peano5nni Unicode version

Theorem peano5nni 8691
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  e.  A  /\  A. x  e.  A  ( x  +  1 )  e.  A )  ->  NN  C_  A )
Distinct variable group:    x, A

Proof of Theorem peano5nni
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 1re 7733 . . . 4  |-  1  e.  RR
2 elin 3229 . . . . 5  |-  ( 1  e.  ( A  i^i  RR )  <->  ( 1  e.  A  /\  1  e.  RR ) )
32biimpri 132 . . . 4  |-  ( ( 1  e.  A  /\  1  e.  RR )  ->  1  e.  ( A  i^i  RR ) )
41, 3mpan2 421 . . 3  |-  ( 1  e.  A  ->  1  e.  ( A  i^i  RR ) )
5 inss1 3266 . . . . 5  |-  ( A  i^i  RR )  C_  A
6 ssralv 3131 . . . . 5  |-  ( ( A  i^i  RR ) 
C_  A  ->  ( A. x  e.  A  ( x  +  1
)  e.  A  ->  A. x  e.  ( A  i^i  RR ) ( x  +  1 )  e.  A ) )
75, 6ax-mp 5 . . . 4  |-  ( A. x  e.  A  (
x  +  1 )  e.  A  ->  A. x  e.  ( A  i^i  RR ) ( x  + 
1 )  e.  A
)
8 inss2 3267 . . . . . . . 8  |-  ( A  i^i  RR )  C_  RR
98sseli 3063 . . . . . . 7  |-  ( x  e.  ( A  i^i  RR )  ->  x  e.  RR )
10 1red 7749 . . . . . . 7  |-  ( x  e.  ( A  i^i  RR )  ->  1  e.  RR )
119, 10readdcld 7763 . . . . . 6  |-  ( x  e.  ( A  i^i  RR )  ->  ( x  +  1 )  e.  RR )
12 elin 3229 . . . . . . 7  |-  ( ( x  +  1 )  e.  ( A  i^i  RR )  <->  ( ( x  +  1 )  e.  A  /\  ( x  +  1 )  e.  RR ) )
1312simplbi2com 1405 . . . . . 6  |-  ( ( x  +  1 )  e.  RR  ->  (
( x  +  1 )  e.  A  -> 
( x  +  1 )  e.  ( A  i^i  RR ) ) )
1411, 13syl 14 . . . . 5  |-  ( x  e.  ( A  i^i  RR )  ->  ( (
x  +  1 )  e.  A  ->  (
x  +  1 )  e.  ( A  i^i  RR ) ) )
1514ralimia 2470 . . . 4  |-  ( A. x  e.  ( A  i^i  RR ) ( x  +  1 )  e.  A  ->  A. x  e.  ( A  i^i  RR ) ( x  + 
1 )  e.  ( A  i^i  RR ) )
167, 15syl 14 . . 3  |-  ( A. x  e.  A  (
x  +  1 )  e.  A  ->  A. x  e.  ( A  i^i  RR ) ( x  + 
1 )  e.  ( A  i^i  RR ) )
17 reex 7722 . . . . 5  |-  RR  e.  _V
1817inex2 4033 . . . 4  |-  ( A  i^i  RR )  e. 
_V
19 eleq2 2181 . . . . . . 7  |-  ( y  =  ( A  i^i  RR )  ->  ( 1  e.  y  <->  1  e.  ( A  i^i  RR ) ) )
20 eleq2 2181 . . . . . . . 8  |-  ( y  =  ( A  i^i  RR )  ->  ( (
x  +  1 )  e.  y  <->  ( x  +  1 )  e.  ( A  i^i  RR ) ) )
2120raleqbi1dv 2611 . . . . . . 7  |-  ( y  =  ( A  i^i  RR )  ->  ( A. x  e.  y  (
x  +  1 )  e.  y  <->  A. x  e.  ( A  i^i  RR ) ( x  + 
1 )  e.  ( A  i^i  RR ) ) )
2219, 21anbi12d 464 . . . . . 6  |-  ( y  =  ( A  i^i  RR )  ->  ( (
1  e.  y  /\  A. x  e.  y  ( x  +  1 )  e.  y )  <->  ( 1  e.  ( A  i^i  RR )  /\  A. x  e.  ( A  i^i  RR ) ( x  + 
1 )  e.  ( A  i^i  RR ) ) ) )
2322elabg 2803 . . . . 5  |-  ( ( A  i^i  RR )  e.  _V  ->  (
( A  i^i  RR )  e.  { y  |  ( 1  e.  y  /\  A. x  e.  y  ( x  +  1 )  e.  y ) }  <->  ( 1  e.  ( A  i^i  RR )  /\  A. x  e.  ( A  i^i  RR ) ( x  + 
1 )  e.  ( A  i^i  RR ) ) ) )
24 dfnn2 8690 . . . . . 6  |-  NN  =  |^| { y  |  ( 1  e.  y  /\  A. x  e.  y  ( x  +  1 )  e.  y ) }
25 intss1 3756 . . . . . 6  |-  ( ( A  i^i  RR )  e.  { y  |  ( 1  e.  y  /\  A. x  e.  y  ( x  + 
1 )  e.  y ) }  ->  |^| { y  |  ( 1  e.  y  /\  A. x  e.  y  ( x  +  1 )  e.  y ) }  C_  ( A  i^i  RR ) )
2624, 25eqsstrid 3113 . . . . 5  |-  ( ( A  i^i  RR )  e.  { y  |  ( 1  e.  y  /\  A. x  e.  y  ( x  + 
1 )  e.  y ) }  ->  NN  C_  ( A  i^i  RR ) )
2723, 26syl6bir 163 . . . 4  |-  ( ( A  i^i  RR )  e.  _V  ->  (
( 1  e.  ( A  i^i  RR )  /\  A. x  e.  ( A  i^i  RR ) ( x  + 
1 )  e.  ( A  i^i  RR ) )  ->  NN  C_  ( A  i^i  RR ) ) )
2818, 27ax-mp 5 . . 3  |-  ( ( 1  e.  ( A  i^i  RR )  /\  A. x  e.  ( A  i^i  RR ) ( x  +  1 )  e.  ( A  i^i  RR ) )  ->  NN  C_  ( A  i^i  RR ) )
294, 16, 28syl2an 287 . 2  |-  ( ( 1  e.  A  /\  A. x  e.  A  ( x  +  1 )  e.  A )  ->  NN  C_  ( A  i^i  RR ) )
3029, 5sstrdi 3079 1  |-  ( ( 1  e.  A  /\  A. x  e.  A  ( x  +  1 )  e.  A )  ->  NN  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   {cab 2103   A.wral 2393   _Vcvv 2660    i^i cin 3040    C_ wss 3041   |^|cint 3741  (class class class)co 5742   RRcr 7587   1c1 7589    + caddc 7591   NNcn 8688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-in 3047  df-ss 3054  df-int 3742  df-inn 8689
This theorem is referenced by:  nnssre  8692  nnind  8704
  Copyright terms: Public domain W3C validator