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Theorem findcard 7113
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 6901 . . 3  |-  ( x  e.  Fin  <->  E. w  e.  om  x  ~~  w
)
3 breq2 4043 . . . . . . . 8  |-  ( w  =  (/)  ->  ( x 
~~  w  <->  x  ~~  (/) ) )
43imbi1d 308 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( x  ~~  w  ->  ph )  <->  ( x  ~~  (/) 
->  ph ) ) )
54albidv 1615 . . . . . 6  |-  ( w  =  (/)  ->  ( A. x ( x  ~~  w  ->  ph )  <->  A. x
( x  ~~  (/)  ->  ph )
) )
6 breq2 4043 . . . . . . . 8  |-  ( w  =  v  ->  (
x  ~~  w  <->  x  ~~  v ) )
76imbi1d 308 . . . . . . 7  |-  ( w  =  v  ->  (
( x  ~~  w  ->  ph )  <->  ( x  ~~  v  ->  ph )
) )
87albidv 1615 . . . . . 6  |-  ( w  =  v  ->  ( A. x ( x  ~~  w  ->  ph )  <->  A. x
( x  ~~  v  ->  ph ) ) )
9 breq2 4043 . . . . . . . 8  |-  ( w  =  suc  v  -> 
( x  ~~  w  <->  x 
~~  suc  v )
)
109imbi1d 308 . . . . . . 7  |-  ( w  =  suc  v  -> 
( ( x  ~~  w  ->  ph )  <->  ( x  ~~  suc  v  ->  ph )
) )
1110albidv 1615 . . . . . 6  |-  ( w  =  suc  v  -> 
( A. x ( x  ~~  w  ->  ph )  <->  A. x ( x 
~~  suc  v  ->  ph ) ) )
12 en0 6940 . . . . . . . 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 1536 . . . . . 6  |-  A. x
( x  ~~  (/)  ->  ph )
18 peano2 4692 . . . . . . . . . . . . 13  |-  ( v  e.  om  ->  suc  v  e.  om )
19 breq2 4043 . . . . . . . . . . . . . 14  |-  ( w  =  suc  v  -> 
( y  ~~  w  <->  y 
~~  suc  v )
)
2019rspcev 2897 . . . . . . . . . . . . 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 6901 . . . . . . . . . . . 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 7107 . . . . . . . . . . . . . . . 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 2804 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
28 difexg 4178 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  _V  ->  (
y  \  { z } )  e.  _V )
2927, 28ax-mp 8 . . . . . . . . . . . . . . . 16  |-  ( y 
\  { z } )  e.  _V
30 breq1 4042 . . . . . . . . . . . . . . . . 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 2887 . . . . . . . . . . . . . . 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 2646 . . . . . . . . . . . . 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 1623 . . . . . . 7  |-  ( v  e.  om  ->  ( A. x ( x  ~~  v  ->  ph )  ->  A. y
( y  ~~  suc  v  ->  th ) ) )
43 breq1 4042 . . . . . . . . 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 1955 . . . . . . 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 4701 . . . . 5  |-  ( w  e.  om  ->  A. x
( x  ~~  w  ->  ph ) )
494819.21bi 1806 . . . 4  |-  ( w  e.  om  ->  (
x  ~~  w  ->  ph ) )
5049rexlimiv 2674 . . 3  |-  ( E. w  e.  om  x  ~~  w  ->  ph )
512, 50sylbi 187 . 2  |-  ( x  e.  Fin  ->  ph )
521, 51vtoclga 2862 1  |-  ( A  e.  Fin  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1530    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    \ cdif 3162   (/)c0 3468   {csn 3653   class class class wbr 4039   suc csuc 4410   omcom 4672    ~~ cen 6876   Fincfn 6879
This theorem is referenced by:  xpfi  7144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-er 6676  df-en 6880  df-fin 6883
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