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Theorem peano5nni 8098
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 7169 . . . 4  |-  1  e.  RR
2 elin 3156 . . . . 5  |-  ( 1  e.  ( A  i^i  RR )  <->  ( 1  e.  A  /\  1  e.  RR ) )
32biimpri 131 . . . 4  |-  ( ( 1  e.  A  /\  1  e.  RR )  ->  1  e.  ( A  i^i  RR ) )
41, 3mpan2 416 . . 3  |-  ( 1  e.  A  ->  1  e.  ( A  i^i  RR ) )
5 inss1 3187 . . . . 5  |-  ( A  i^i  RR )  C_  A
6 ssralv 3059 . . . . 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 3188 . . . . . . . 8  |-  ( A  i^i  RR )  C_  RR
98sseli 2996 . . . . . . 7  |-  ( x  e.  ( A  i^i  RR )  ->  x  e.  RR )
10 1red 7185 . . . . . . 7  |-  ( x  e.  ( A  i^i  RR )  ->  1  e.  RR )
119, 10readdcld 7199 . . . . . 6  |-  ( x  e.  ( A  i^i  RR )  ->  ( x  +  1 )  e.  RR )
12 elin 3156 . . . . . . 7  |-  ( ( x  +  1 )  e.  ( A  i^i  RR )  <->  ( ( x  +  1 )  e.  A  /\  ( x  +  1 )  e.  RR ) )
1312simplbi2com 1374 . . . . . 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 2425 . . . 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 7158 . . . . 5  |-  RR  e.  _V
1817inex2 3915 . . . 4  |-  ( A  i^i  RR )  e. 
_V
19 eleq2 2143 . . . . . . 7  |-  ( y  =  ( A  i^i  RR )  ->  ( 1  e.  y  <->  1  e.  ( A  i^i  RR ) ) )
20 eleq2 2143 . . . . . . . 8  |-  ( y  =  ( A  i^i  RR )  ->  ( (
x  +  1 )  e.  y  <->  ( x  +  1 )  e.  ( A  i^i  RR ) ) )
2120raleqbi1dv 2558 . . . . . . 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 457 . . . . . 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 2740 . . . . 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 8097 . . . . . 6  |-  NN  =  |^| { y  |  ( 1  e.  y  /\  A. x  e.  y  ( x  +  1 )  e.  y ) }
25 intss1 3653 . . . . . 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, 25syl5eqss 3044 . . . . 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 162 . . . 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 283 . 2  |-  ( ( 1  e.  A  /\  A. x  e.  A  ( x  +  1 )  e.  A )  ->  NN  C_  ( A  i^i  RR ) )
3029, 5syl6ss 3012 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 102    = wceq 1285    e. wcel 1434   {cab 2068   A.wral 2349   _Vcvv 2602    i^i cin 2973    C_ wss 2974   |^|cint 3638  (class class class)co 5537   RRcr 7031   1c1 7033    + caddc 7035   NNcn 8095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-cnex 7118  ax-resscn 7119  ax-1re 7121  ax-addrcl 7124
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-in 2980  df-ss 2987  df-int 3639  df-inn 8096
This theorem is referenced by:  nnssre  8099  nnind  8111
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