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Theorem nnind 9911
Description: Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. See nnaddcl 9915 for an example of its use. See nn0ind 10259 for induction on nonnegative integers and uzind 10254, uzind4 10427 for induction on an arbitrary set of upper integers. See indstr 10438 for strong induction. See also nnindALT 9912. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
Hypotheses
Ref Expression
nnind.1  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
nnind.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
nnind.3  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
nnind.4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
nnind.5  |-  ps
nnind.6  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
Assertion
Ref Expression
nnind  |-  ( A  e.  NN  ->  ta )
Distinct variable groups:    x, y    x, A    ps, x    ch, x    th, x    ta, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)    ch( y)    th( y)    ta( y)    A( y)

Proof of Theorem nnind
StepHypRef Expression
1 1nn 9904 . . . . . 6  |-  1  e.  NN
2 nnind.5 . . . . . 6  |-  ps
3 nnind.1 . . . . . . 7  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
43elrab 3009 . . . . . 6  |-  ( 1  e.  { x  e.  NN  |  ph }  <->  ( 1  e.  NN  /\  ps ) )
51, 2, 4mpbir2an 886 . . . . 5  |-  1  e.  { x  e.  NN  |  ph }
6 ssrab2 3344 . . . . . . . 8  |-  { x  e.  NN  |  ph }  C_  NN
76sseli 3262 . . . . . . 7  |-  ( y  e.  { x  e.  NN  |  ph }  ->  y  e.  NN )
8 peano2nn 9905 . . . . . . . . . 10  |-  ( y  e.  NN  ->  (
y  +  1 )  e.  NN )
98a1d 22 . . . . . . . . 9  |-  ( y  e.  NN  ->  (
y  e.  NN  ->  ( y  +  1 )  e.  NN ) )
10 nnind.6 . . . . . . . . 9  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
119, 10anim12d 546 . . . . . . . 8  |-  ( y  e.  NN  ->  (
( y  e.  NN  /\ 
ch )  ->  (
( y  +  1 )  e.  NN  /\  th ) ) )
12 nnind.2 . . . . . . . . 9  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
1312elrab 3009 . . . . . . . 8  |-  ( y  e.  { x  e.  NN  |  ph }  <->  ( y  e.  NN  /\  ch ) )
14 nnind.3 . . . . . . . . 9  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
1514elrab 3009 . . . . . . . 8  |-  ( ( y  +  1 )  e.  { x  e.  NN  |  ph }  <->  ( ( y  +  1 )  e.  NN  /\  th ) )
1611, 13, 153imtr4g 261 . . . . . . 7  |-  ( y  e.  NN  ->  (
y  e.  { x  e.  NN  |  ph }  ->  ( y  +  1 )  e.  { x  e.  NN  |  ph }
) )
177, 16mpcom 32 . . . . . 6  |-  ( y  e.  { x  e.  NN  |  ph }  ->  ( y  +  1 )  e.  { x  e.  NN  |  ph }
)
1817rgen 2693 . . . . 5  |-  A. y  e.  { x  e.  NN  |  ph }  ( y  +  1 )  e. 
{ x  e.  NN  |  ph }
19 peano5nni 9896 . . . . 5  |-  ( ( 1  e.  { x  e.  NN  |  ph }  /\  A. y  e.  {
x  e.  NN  |  ph }  ( y  +  1 )  e.  {
x  e.  NN  |  ph } )  ->  NN  C_ 
{ x  e.  NN  |  ph } )
205, 18, 19mp2an 653 . . . 4  |-  NN  C_  { x  e.  NN  |  ph }
2120sseli 3262 . . 3  |-  ( A  e.  NN  ->  A  e.  { x  e.  NN  |  ph } )
22 nnind.4 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
2322elrab 3009 . . 3  |-  ( A  e.  { x  e.  NN  |  ph }  <->  ( A  e.  NN  /\  ta ) )
2421, 23sylib 188 . 2  |-  ( A  e.  NN  ->  ( A  e.  NN  /\  ta ) )
2524simprd 449 1  |-  ( A  e.  NN  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   A.wral 2628   {crab 2632    C_ wss 3238  (class class class)co 5981   1c1 8885    + caddc 8887   NNcn 9893
This theorem is referenced by:  nnindALT  9912  nn1m1nn  9913  nnaddcl  9915  nnmulcl  9916  nnge1  9919  nnsub  9931  nneo  10246  peano5uzi  10251  uzindOLD  10257  nn0ind-raph  10263  ser1const  11266  expcllem  11279  expeq0  11297  seqcoll  11599  climcndslem2  12517  sqr2irr  12735  gcdmultiple  12937  rplpwr  12943  prmind2  12977  prmdvdsexp  13001  eulerthlem2  13058  pcmpt  13148  prmpwdvds  13159  vdwlem10  13245  mulgnnass  14805  imasdsf1olem  18150  ovolunlem1a  19070  ovolicc2lem3  19093  voliunlem1  19122  volsup  19128  dvexp  19517  plyco  19838  dgrcolem1  19869  vieta1  19907  emcllem6  20517  bposlem5  20750  2sqlem10  20836  dchrisum0flb  20882  iuninc  23409  esumfzf  23924  rrvsum  24160  subfacp1lem6  24319  cvmliftlem10  24428  gammacvglem1  24834  faclimlem1  24837  incsequz  25965  bfplem1  26052  2nn0ind  26536  expmordi  26538  fmuldfeq  27219  stoweidlem20  27275  wallispilem4  27323  wallispi2lem1  27326  wallispi2lem2  27327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-1cn 8942
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-recs 6530  df-rdg 6565  df-nn 9894
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