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Theorem peano5nni 8941
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 7975 . . . 4  |-  1  e.  RR
2 elin 3333 . . . . 5  |-  ( 1  e.  ( A  i^i  RR )  <->  ( 1  e.  A  /\  1  e.  RR ) )
32biimpri 133 . . . 4  |-  ( ( 1  e.  A  /\  1  e.  RR )  ->  1  e.  ( A  i^i  RR ) )
41, 3mpan2 425 . . 3  |-  ( 1  e.  A  ->  1  e.  ( A  i^i  RR ) )
5 inss1 3370 . . . . 5  |-  ( A  i^i  RR )  C_  A
6 ssralv 3234 . . . . 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 3371 . . . . . . . 8  |-  ( A  i^i  RR )  C_  RR
98sseli 3166 . . . . . . 7  |-  ( x  e.  ( A  i^i  RR )  ->  x  e.  RR )
10 1red 7991 . . . . . . 7  |-  ( x  e.  ( A  i^i  RR )  ->  1  e.  RR )
119, 10readdcld 8006 . . . . . 6  |-  ( x  e.  ( A  i^i  RR )  ->  ( x  +  1 )  e.  RR )
12 elin 3333 . . . . . . 7  |-  ( ( x  +  1 )  e.  ( A  i^i  RR )  <->  ( ( x  +  1 )  e.  A  /\  ( x  +  1 )  e.  RR ) )
1312simplbi2com 1455 . . . . . 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 2551 . . . 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 7964 . . . . 5  |-  RR  e.  _V
1817inex2 4153 . . . 4  |-  ( A  i^i  RR )  e. 
_V
19 eleq2 2253 . . . . . . 7  |-  ( y  =  ( A  i^i  RR )  ->  ( 1  e.  y  <->  1  e.  ( A  i^i  RR ) ) )
20 eleq2 2253 . . . . . . . 8  |-  ( y  =  ( A  i^i  RR )  ->  ( (
x  +  1 )  e.  y  <->  ( x  +  1 )  e.  ( A  i^i  RR ) ) )
2120raleqbi1dv 2694 . . . . . . 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 473 . . . . . 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 2898 . . . . 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 8940 . . . . . 6  |-  NN  =  |^| { y  |  ( 1  e.  y  /\  A. x  e.  y  ( x  +  1 )  e.  y ) }
25 intss1 3874 . . . . . 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 3216 . . . . 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 164 . . . 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 289 . 2  |-  ( ( 1  e.  A  /\  A. x  e.  A  ( x  +  1 )  e.  A )  ->  NN  C_  ( A  i^i  RR ) )
3029, 5sstrdi 3182 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 104    = wceq 1364    e. wcel 2160   {cab 2175   A.wral 2468   _Vcvv 2752    i^i cin 3143    C_ wss 3144   |^|cint 3859  (class class class)co 5891   RRcr 7829   1c1 7831    + caddc 7833   NNcn 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4136  ax-cnex 7921  ax-resscn 7922  ax-1re 7924  ax-addrcl 7927
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-int 3860  df-inn 8939
This theorem is referenced by:  nnssre  8942  nnind  8954
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