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Theorem findcard 7339
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 7123 . . 3  |-  ( x  e.  Fin  <->  E. w  e.  om  x  ~~  w
)
3 breq2 4208 . . . . . . . 8  |-  ( w  =  (/)  ->  ( x 
~~  w  <->  x  ~~  (/) ) )
43imbi1d 309 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( x  ~~  w  ->  ph )  <->  ( x  ~~  (/) 
->  ph ) ) )
54albidv 1635 . . . . . 6  |-  ( w  =  (/)  ->  ( A. x ( x  ~~  w  ->  ph )  <->  A. x
( x  ~~  (/)  ->  ph )
) )
6 breq2 4208 . . . . . . . 8  |-  ( w  =  v  ->  (
x  ~~  w  <->  x  ~~  v ) )
76imbi1d 309 . . . . . . 7  |-  ( w  =  v  ->  (
( x  ~~  w  ->  ph )  <->  ( x  ~~  v  ->  ph )
) )
87albidv 1635 . . . . . 6  |-  ( w  =  v  ->  ( A. x ( x  ~~  w  ->  ph )  <->  A. x
( x  ~~  v  ->  ph ) ) )
9 breq2 4208 . . . . . . . 8  |-  ( w  =  suc  v  -> 
( x  ~~  w  <->  x 
~~  suc  v )
)
109imbi1d 309 . . . . . . 7  |-  ( w  =  suc  v  -> 
( ( x  ~~  w  ->  ph )  <->  ( x  ~~  suc  v  ->  ph )
) )
1110albidv 1635 . . . . . 6  |-  ( w  =  suc  v  -> 
( A. x ( x  ~~  w  ->  ph )  <->  A. x ( x 
~~  suc  v  ->  ph ) ) )
12 en0 7162 . . . . . . . 8  |-  ( x 
~~  (/)  <->  x  =  (/) )
13 findcard.5 . . . . . . . . 9  |-  ps
14 findcard.1 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
1513, 14mpbiri 225 . . . . . . . 8  |-  ( x  =  (/)  ->  ph )
1612, 15sylbi 188 . . . . . . 7  |-  ( x 
~~  (/)  ->  ph )
1716ax-gen 1555 . . . . . 6  |-  A. x
( x  ~~  (/)  ->  ph )
18 peano2 4857 . . . . . . . . . . . . 13  |-  ( v  e.  om  ->  suc  v  e.  om )
19 breq2 4208 . . . . . . . . . . . . . 14  |-  ( w  =  suc  v  -> 
( y  ~~  w  <->  y 
~~  suc  v )
)
2019rspcev 3044 . . . . . . . . . . . . 13  |-  ( ( suc  v  e.  om  /\  y  ~~  suc  v
)  ->  E. w  e.  om  y  ~~  w
)
2118, 20sylan 458 . . . . . . . . . . . 12  |-  ( ( v  e.  om  /\  y  ~~  suc  v )  ->  E. w  e.  om  y  ~~  w )
22 isfi 7123 . . . . . . . . . . . 12  |-  ( y  e.  Fin  <->  E. w  e.  om  y  ~~  w
)
2321, 22sylibr 204 . . . . . . . . . . 11  |-  ( ( v  e.  om  /\  y  ~~  suc  v )  ->  y  e.  Fin )
24233adant2 976 . . . . . . . . . 10  |-  ( ( v  e.  om  /\  A. x ( x  ~~  v  ->  ph )  /\  y  ~~  suc  v )  -> 
y  e.  Fin )
25 dif1en 7333 . . . . . . . . . . . . . . . 16  |-  ( ( v  e.  om  /\  y  ~~  suc  v  /\  z  e.  y )  ->  ( y  \  {
z } )  ~~  v )
26253expa 1153 . . . . . . . . . . . . . . 15  |-  ( ( ( v  e.  om  /\  y  ~~  suc  v
)  /\  z  e.  y )  ->  (
y  \  { z } )  ~~  v
)
27 vex 2951 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
28 difexg 4343 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  _V  ->  (
y  \  { z } )  e.  _V )
2927, 28ax-mp 8 . . . . . . . . . . . . . . . 16  |-  ( y 
\  { z } )  e.  _V
30 breq1 4207 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  \  { z } )  ->  ( x  ~~  v 
<->  ( y  \  {
z } )  ~~  v ) )
31 findcard.2 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  \  { z } )  ->  ( ph  <->  ch )
)
3230, 31imbi12d 312 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( y  \  { z } )  ->  ( ( x 
~~  v  ->  ph )  <->  ( ( y  \  {
z } )  ~~  v  ->  ch ) ) )
3329, 32spcv 3034 . . . . . . . . . . . . . . 15  |-  ( A. x ( x  ~~  v  ->  ph )  ->  (
( y  \  {
z } )  ~~  v  ->  ch ) )
3426, 33syl5com 28 . . . . . . . . . . . . . 14  |-  ( ( ( v  e.  om  /\  y  ~~  suc  v
)  /\  z  e.  y )  ->  ( A. x ( x  ~~  v  ->  ph )  ->  ch ) )
3534ralrimdva 2788 . . . . . . . . . . . . 13  |-  ( ( v  e.  om  /\  y  ~~  suc  v )  ->  ( A. x
( x  ~~  v  ->  ph )  ->  A. z  e.  y  ch )
)
3635imp 419 . . . . . . . . . . . 12  |-  ( ( ( v  e.  om  /\  y  ~~  suc  v
)  /\  A. x
( x  ~~  v  ->  ph ) )  ->  A. z  e.  y  ch )
3736an32s 780 . . . . . . . . . . 11  |-  ( ( ( v  e.  om  /\ 
A. x ( x 
~~  v  ->  ph )
)  /\  y  ~~  suc  v )  ->  A. z  e.  y  ch )
38373impa 1148 . . . . . . . . . 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 58 . . . . . . . . 9  |-  ( ( v  e.  om  /\  A. x ( x  ~~  v  ->  ph )  /\  y  ~~  suc  v )  ->  th )
41403exp 1152 . . . . . . . 8  |-  ( v  e.  om  ->  ( A. x ( x  ~~  v  ->  ph )  ->  (
y  ~~  suc  v  ->  th ) ) )
4241alrimdv 1643 . . . . . . 7  |-  ( v  e.  om  ->  ( A. x ( x  ~~  v  ->  ph )  ->  A. y
( y  ~~  suc  v  ->  th ) ) )
43 breq1 4207 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  ~~  suc  v  <->  y  ~~  suc  v ) )
44 findcard.3 . . . . . . . . 9  |-  ( x  =  y  ->  ( ph 
<->  th ) )
4543, 44imbi12d 312 . . . . . . . 8  |-  ( x  =  y  ->  (
( x  ~~  suc  v  ->  ph )  <->  ( y  ~~  suc  v  ->  th )
) )
4645cbvalv 1984 . . . . . . 7  |-  ( A. x ( x  ~~  suc  v  ->  ph )  <->  A. y ( y  ~~  suc  v  ->  th )
)
4742, 46syl6ibr 219 . . . . . 6  |-  ( v  e.  om  ->  ( A. x ( x  ~~  v  ->  ph )  ->  A. x
( x  ~~  suc  v  ->  ph ) ) )
485, 8, 11, 17, 47finds1 4866 . . . . 5  |-  ( w  e.  om  ->  A. x
( x  ~~  w  ->  ph ) )
494819.21bi 1774 . . . 4  |-  ( w  e.  om  ->  (
x  ~~  w  ->  ph ) )
5049rexlimiv 2816 . . 3  |-  ( E. w  e.  om  x  ~~  w  ->  ph )
512, 50sylbi 188 . 2  |-  ( x  e.  Fin  ->  ph )
521, 51vtoclga 3009 1  |-  ( A  e.  Fin  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948    \ cdif 3309   (/)c0 3620   {csn 3806   class class class wbr 4204   suc csuc 4575   omcom 4837    ~~ cen 7098   Fincfn 7101
This theorem is referenced by:  xpfi  7370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1o 6716  df-er 6897  df-en 7102  df-fin 7105
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