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Theorem findcard 7097
Description: Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
findcard.1  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
findcard.2  |-  ( x  =  ( y  \  { z } )  ->  ( ph  <->  ch )
)
findcard.3  |-  ( x  =  y  ->  ( ph 
<->  th ) )
findcard.4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
findcard.5  |-  ps
findcard.6  |-  ( y  e.  Fin  ->  ( A. z  e.  y  ch  ->  th ) )
Assertion
Ref Expression
findcard  |-  ( A  e.  Fin  ->  ta )
Distinct variable groups:    x, y,
z, A    ps, x    ch, x    th, x    ta, x    ph, y, z
Allowed substitution hints:    ph( x)    ps( y, z)    ch( y, z)    th( y, z)    ta( y,
z)

Proof of Theorem findcard
Dummy variables  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 findcard.4 . 2  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
2 isfi 6885 . . 3  |-  ( x  e.  Fin  <->  E. w  e.  om  x  ~~  w
)
3 breq2 4027 . . . . . . . 8  |-  ( w  =  (/)  ->  ( x 
~~  w  <->  x  ~~  (/) ) )
43imbi1d 308 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( x  ~~  w  ->  ph )  <->  ( x  ~~  (/) 
->  ph ) ) )
54albidv 1611 . . . . . 6  |-  ( w  =  (/)  ->  ( A. x ( x  ~~  w  ->  ph )  <->  A. x
( x  ~~  (/)  ->  ph )
) )
6 breq2 4027 . . . . . . . 8  |-  ( w  =  v  ->  (
x  ~~  w  <->  x  ~~  v ) )
76imbi1d 308 . . . . . . 7  |-  ( w  =  v  ->  (
( x  ~~  w  ->  ph )  <->  ( x  ~~  v  ->  ph )
) )
87albidv 1611 . . . . . 6  |-  ( w  =  v  ->  ( A. x ( x  ~~  w  ->  ph )  <->  A. x
( x  ~~  v  ->  ph ) ) )
9 breq2 4027 . . . . . . . 8  |-  ( w  =  suc  v  -> 
( x  ~~  w  <->  x 
~~  suc  v )
)
109imbi1d 308 . . . . . . 7  |-  ( w  =  suc  v  -> 
( ( x  ~~  w  ->  ph )  <->  ( x  ~~  suc  v  ->  ph )
) )
1110albidv 1611 . . . . . 6  |-  ( w  =  suc  v  -> 
( A. x ( x  ~~  w  ->  ph )  <->  A. x ( x 
~~  suc  v  ->  ph ) ) )
12 en0 6924 . . . . . . . 8  |-  ( x 
~~  (/)  <->  x  =  (/) )
13 findcard.5 . . . . . . . . 9  |-  ps
14 findcard.1 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
1513, 14mpbiri 224 . . . . . . . 8  |-  ( x  =  (/)  ->  ph )
1612, 15sylbi 187 . . . . . . 7  |-  ( x 
~~  (/)  ->  ph )
1716ax-gen 1533 . . . . . 6  |-  A. x
( x  ~~  (/)  ->  ph )
18 peano2 4676 . . . . . . . . . . . . 13  |-  ( v  e.  om  ->  suc  v  e.  om )
19 breq2 4027 . . . . . . . . . . . . . 14  |-  ( w  =  suc  v  -> 
( y  ~~  w  <->  y 
~~  suc  v )
)
2019rspcev 2884 . . . . . . . . . . . . 13  |-  ( ( suc  v  e.  om  /\  y  ~~  suc  v
)  ->  E. w  e.  om  y  ~~  w
)
2118, 20sylan 457 . . . . . . . . . . . 12  |-  ( ( v  e.  om  /\  y  ~~  suc  v )  ->  E. w  e.  om  y  ~~  w )
22 isfi 6885 . . . . . . . . . . . 12  |-  ( y  e.  Fin  <->  E. w  e.  om  y  ~~  w
)
2321, 22sylibr 203 . . . . . . . . . . 11  |-  ( ( v  e.  om  /\  y  ~~  suc  v )  ->  y  e.  Fin )
24233adant2 974 . . . . . . . . . 10  |-  ( ( v  e.  om  /\  A. x ( x  ~~  v  ->  ph )  /\  y  ~~  suc  v )  -> 
y  e.  Fin )
25 dif1en 7091 . . . . . . . . . . . . . . . 16  |-  ( ( v  e.  om  /\  y  ~~  suc  v  /\  z  e.  y )  ->  ( y  \  {
z } )  ~~  v )
26253expa 1151 . . . . . . . . . . . . . . 15  |-  ( ( ( v  e.  om  /\  y  ~~  suc  v
)  /\  z  e.  y )  ->  (
y  \  { z } )  ~~  v
)
27 vex 2791 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
28 difexg 4162 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  _V  ->  (
y  \  { z } )  e.  _V )
2927, 28ax-mp 8 . . . . . . . . . . . . . . . 16  |-  ( y 
\  { z } )  e.  _V
30 breq1 4026 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  \  { z } )  ->  ( x  ~~  v 
<->  ( y  \  {
z } )  ~~  v ) )
31 findcard.2 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  \  { z } )  ->  ( ph  <->  ch )
)
3230, 31imbi12d 311 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( y  \  { z } )  ->  ( ( x 
~~  v  ->  ph )  <->  ( ( y  \  {
z } )  ~~  v  ->  ch ) ) )
3329, 32spcv 2874 . . . . . . . . . . . . . . 15  |-  ( A. x ( x  ~~  v  ->  ph )  ->  (
( y  \  {
z } )  ~~  v  ->  ch ) )
3426, 33syl5com 26 . . . . . . . . . . . . . 14  |-  ( ( ( v  e.  om  /\  y  ~~  suc  v
)  /\  z  e.  y )  ->  ( A. x ( x  ~~  v  ->  ph )  ->  ch ) )
3534ralrimdva 2633 . . . . . . . . . . . . 13  |-  ( ( v  e.  om  /\  y  ~~  suc  v )  ->  ( A. x
( x  ~~  v  ->  ph )  ->  A. z  e.  y  ch )
)
3635imp 418 . . . . . . . . . . . 12  |-  ( ( ( v  e.  om  /\  y  ~~  suc  v
)  /\  A. x
( x  ~~  v  ->  ph ) )  ->  A. z  e.  y  ch )
3736an32s 779 . . . . . . . . . . 11  |-  ( ( ( v  e.  om  /\ 
A. x ( x 
~~  v  ->  ph )
)  /\  y  ~~  suc  v )  ->  A. z  e.  y  ch )
38373impa 1146 . . . . . . . . . 10  |-  ( ( v  e.  om  /\  A. x ( x  ~~  v  ->  ph )  /\  y  ~~  suc  v )  ->  A. z  e.  y  ch )
39 findcard.6 . . . . . . . . . 10  |-  ( y  e.  Fin  ->  ( A. z  e.  y  ch  ->  th ) )
4024, 38, 39sylc 56 . . . . . . . . 9  |-  ( ( v  e.  om  /\  A. x ( x  ~~  v  ->  ph )  /\  y  ~~  suc  v )  ->  th )
41403exp 1150 . . . . . . . 8  |-  ( v  e.  om  ->  ( A. x ( x  ~~  v  ->  ph )  ->  (
y  ~~  suc  v  ->  th ) ) )
4241alrimdv 1619 . . . . . . 7  |-  ( v  e.  om  ->  ( A. x ( x  ~~  v  ->  ph )  ->  A. y
( y  ~~  suc  v  ->  th ) ) )
43 breq1 4026 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  ~~  suc  v  <->  y  ~~  suc  v ) )
44 findcard.3 . . . . . . . . 9  |-  ( x  =  y  ->  ( ph 
<->  th ) )
4543, 44imbi12d 311 . . . . . . . 8  |-  ( x  =  y  ->  (
( x  ~~  suc  v  ->  ph )  <->  ( y  ~~  suc  v  ->  th )
) )
4645cbvalv 1942 . . . . . . 7  |-  ( A. x ( x  ~~  suc  v  ->  ph )  <->  A. y ( y  ~~  suc  v  ->  th )
)
4742, 46syl6ibr 218 . . . . . 6  |-  ( v  e.  om  ->  ( A. x ( x  ~~  v  ->  ph )  ->  A. x
( x  ~~  suc  v  ->  ph ) ) )
485, 8, 11, 17, 47finds1 4685 . . . . 5  |-  ( w  e.  om  ->  A. x
( x  ~~  w  ->  ph ) )
494819.21bi 1794 . . . 4  |-  ( w  e.  om  ->  (
x  ~~  w  ->  ph ) )
5049rexlimiv 2661 . . 3  |-  ( E. w  e.  om  x  ~~  w  ->  ph )
512, 50sylbi 187 . 2  |-  ( x  e.  Fin  ->  ph )
521, 51vtoclga 2849 1  |-  ( A  e.  Fin  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149   (/)c0 3455   {csn 3640   class class class wbr 4023   suc csuc 4394   omcom 4656    ~~ cen 6860   Fincfn 6863
This theorem is referenced by:  xpfi  7128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-er 6660  df-en 6864  df-fin 6867
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