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Theorem nnind 9644
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 9648 for an example of its use. See nn0ind 9987 for induction on nonnegative integers and uzind 9982, uzind4 10155 for induction on an arbitrary set of upper integers. See indstr 10166 for strong induction. (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 9637 . . . . . 6  |-  1  e.  NN
2 nnind.5 . . . . . 6  |-  ps
3 nnind.1 . . . . . . 7  |-  ( x  =  1  ->  ( ph 
<->  ps ) )
43elrab 2860 . . . . . 6  |-  ( 1  e.  { x  e.  NN  |  ph }  <->  ( 1  e.  NN  /\  ps ) )
51, 2, 4mpbir2an 891 . . . . 5  |-  1  e.  { x  e.  NN  |  ph }
6 ssrab2 3179 . . . . . . . 8  |-  { x  e.  NN  |  ph }  C_  NN
76sseli 3099 . . . . . . 7  |-  ( y  e.  { x  e.  NN  |  ph }  ->  y  e.  NN )
8 peano2nn 9638 . . . . . . . . . 10  |-  ( y  e.  NN  ->  (
y  +  1 )  e.  NN )
98a1d 24 . . . . . . . . 9  |-  ( y  e.  NN  ->  (
y  e.  NN  ->  ( y  +  1 )  e.  NN ) )
10 nnind.6 . . . . . . . . 9  |-  ( y  e.  NN  ->  ( ch  ->  th ) )
119, 10anim12d 548 . . . . . . . 8  |-  ( y  e.  NN  ->  (
( y  e.  NN  /\ 
ch )  ->  (
( y  +  1 )  e.  NN  /\  th ) ) )
12 nnind.2 . . . . . . . . 9  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
1312elrab 2860 . . . . . . . 8  |-  ( y  e.  { x  e.  NN  |  ph }  <->  ( y  e.  NN  /\  ch ) )
14 nnind.3 . . . . . . . . 9  |-  ( x  =  ( y  +  1 )  ->  ( ph 
<->  th ) )
1514elrab 2860 . . . . . . . 8  |-  ( ( y  +  1 )  e.  { x  e.  NN  |  ph }  <->  ( ( y  +  1 )  e.  NN  /\  th ) )
1611, 13, 153imtr4g 263 . . . . . . 7  |-  ( y  e.  NN  ->  (
y  e.  { x  e.  NN  |  ph }  ->  ( y  +  1 )  e.  { x  e.  NN  |  ph }
) )
177, 16mpcom 34 . . . . . 6  |-  ( y  e.  { x  e.  NN  |  ph }  ->  ( y  +  1 )  e.  { x  e.  NN  |  ph }
)
1817rgen 2570 . . . . 5  |-  A. y  e.  { x  e.  NN  |  ph }  ( y  +  1 )  e. 
{ x  e.  NN  |  ph }
19 peano5nni 9629 . . . . 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 656 . . . 4  |-  NN  C_  { x  e.  NN  |  ph }
2120sseli 3099 . . 3  |-  ( A  e.  NN  ->  A  e.  { x  e.  NN  |  ph } )
22 nnind.4 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
2322elrab 2860 . . 3  |-  ( A  e.  { x  e.  NN  |  ph }  <->  ( A  e.  NN  /\  ta ) )
2421, 23sylib 190 . 2  |-  ( A  e.  NN  ->  ( A  e.  NN  /\  ta ) )
2524simprd 451 1  |-  ( A  e.  NN  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2509   {crab 2512    C_ wss 3078  (class class class)co 5710   1c1 8618    + caddc 8620   NNcn 9626
This theorem is referenced by:  nnindALT  9645  nn1m1nn  9646  nnaddcl  9648  nnmulcl  9649  nnge1  9652  nnsub  9664  nneo  9974  peano5uzi  9979  uzindOLD  9985  nn0ind-raph  9991  ser1const  10980  expcllem  10992  expeq0  11010  seqcoll  11278  climcndslem2  12183  sqr2irr  12401  gcdmultiple  12603  rplpwr  12609  prmind2  12643  prmdvdsexp  12667  eulerthlem2  12724  pcmpt  12814  prmpwdvds  12825  vdwlem10  12911  mulgnnass  14430  imasdsf1olem  17769  ovolunlem1a  18687  ovolicc2lem3  18710  voliunlem1  18739  volsup  18745  dvexp  19134  plyco  19455  dgrcolem1  19486  vieta1  19524  emcllem6  20126  bposlem5  20359  2sqlem10  20445  dchrisum0flb  20491  subfacp1lem6  22887  cvmliftlem10  22996  incsequz  25624  bfplem1  25712  2nn0ind  26196  expmordi  26198
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403  ax-1cn 8675
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-recs 6274  df-rdg 6309  df-n 9627
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