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Theorem peano5nni 8633
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 7689 . . . 4  |-  1  e.  RR
2 elin 3225 . . . . 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 419 . . 3  |-  ( 1  e.  A  ->  1  e.  ( A  i^i  RR ) )
5 inss1 3262 . . . . 5  |-  ( A  i^i  RR )  C_  A
6 ssralv 3127 . . . . 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 7 . . . 4  |-  ( A. x  e.  A  (
x  +  1 )  e.  A  ->  A. x  e.  ( A  i^i  RR ) ( x  + 
1 )  e.  A
)
8 inss2 3263 . . . . . . . 8  |-  ( A  i^i  RR )  C_  RR
98sseli 3059 . . . . . . 7  |-  ( x  e.  ( A  i^i  RR )  ->  x  e.  RR )
10 1red 7705 . . . . . . 7  |-  ( x  e.  ( A  i^i  RR )  ->  1  e.  RR )
119, 10readdcld 7719 . . . . . 6  |-  ( x  e.  ( A  i^i  RR )  ->  ( x  +  1 )  e.  RR )
12 elin 3225 . . . . . . 7  |-  ( ( x  +  1 )  e.  ( A  i^i  RR )  <->  ( ( x  +  1 )  e.  A  /\  ( x  +  1 )  e.  RR ) )
1312simplbi2com 1403 . . . . . 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 2467 . . . 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 7678 . . . . 5  |-  RR  e.  _V
1817inex2 4023 . . . 4  |-  ( A  i^i  RR )  e. 
_V
19 eleq2 2178 . . . . . . 7  |-  ( y  =  ( A  i^i  RR )  ->  ( 1  e.  y  <->  1  e.  ( A  i^i  RR ) ) )
20 eleq2 2178 . . . . . . . 8  |-  ( y  =  ( A  i^i  RR )  ->  ( (
x  +  1 )  e.  y  <->  ( x  +  1 )  e.  ( A  i^i  RR ) ) )
2120raleqbi1dv 2608 . . . . . . 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 462 . . . . . 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 2799 . . . . 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 8632 . . . . . 6  |-  NN  =  |^| { y  |  ( 1  e.  y  /\  A. x  e.  y  ( x  +  1 )  e.  y ) }
25 intss1 3752 . . . . . 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 3109 . . . . 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 7 . . 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 285 . 2  |-  ( ( 1  e.  A  /\  A. x  e.  A  ( x  +  1 )  e.  A )  ->  NN  C_  ( A  i^i  RR ) )
3029, 5syl6ss 3075 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 1314    e. wcel 1463   {cab 2101   A.wral 2390   _Vcvv 2657    i^i cin 3036    C_ wss 3037   |^|cint 3737  (class class class)co 5728   RRcr 7546   1c1 7548    + caddc 7550   NNcn 8630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-cnex 7636  ax-resscn 7637  ax-1re 7639  ax-addrcl 7642
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-v 2659  df-in 3043  df-ss 3050  df-int 3738  df-inn 8631
This theorem is referenced by:  nnssre  8634  nnind  8646
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