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Theorem findcard 7655
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 7436 . . 3  |-  ( x  e.  Fin  <->  E. w  e.  om  x  ~~  w
)
3 breq2 4397 . . . . . . . 8  |-  ( w  =  (/)  ->  ( x 
~~  w  <->  x  ~~  (/) ) )
43imbi1d 317 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( x  ~~  w  ->  ph )  <->  ( x  ~~  (/) 
->  ph ) ) )
54albidv 1680 . . . . . 6  |-  ( w  =  (/)  ->  ( A. x ( x  ~~  w  ->  ph )  <->  A. x
( x  ~~  (/)  ->  ph )
) )
6 breq2 4397 . . . . . . . 8  |-  ( w  =  v  ->  (
x  ~~  w  <->  x  ~~  v ) )
76imbi1d 317 . . . . . . 7  |-  ( w  =  v  ->  (
( x  ~~  w  ->  ph )  <->  ( x  ~~  v  ->  ph )
) )
87albidv 1680 . . . . . 6  |-  ( w  =  v  ->  ( A. x ( x  ~~  w  ->  ph )  <->  A. x
( x  ~~  v  ->  ph ) ) )
9 breq2 4397 . . . . . . . 8  |-  ( w  =  suc  v  -> 
( x  ~~  w  <->  x 
~~  suc  v )
)
109imbi1d 317 . . . . . . 7  |-  ( w  =  suc  v  -> 
( ( x  ~~  w  ->  ph )  <->  ( x  ~~  suc  v  ->  ph )
) )
1110albidv 1680 . . . . . 6  |-  ( w  =  suc  v  -> 
( A. x ( x  ~~  w  ->  ph )  <->  A. x ( x 
~~  suc  v  ->  ph ) ) )
12 en0 7475 . . . . . . . 8  |-  ( x 
~~  (/)  <->  x  =  (/) )
13 findcard.5 . . . . . . . . 9  |-  ps
14 findcard.1 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
1513, 14mpbiri 233 . . . . . . . 8  |-  ( x  =  (/)  ->  ph )
1612, 15sylbi 195 . . . . . . 7  |-  ( x 
~~  (/)  ->  ph )
1716ax-gen 1592 . . . . . 6  |-  A. x
( x  ~~  (/)  ->  ph )
18 peano2 6599 . . . . . . . . . . . . 13  |-  ( v  e.  om  ->  suc  v  e.  om )
19 breq2 4397 . . . . . . . . . . . . . 14  |-  ( w  =  suc  v  -> 
( y  ~~  w  <->  y 
~~  suc  v )
)
2019rspcev 3172 . . . . . . . . . . . . 13  |-  ( ( suc  v  e.  om  /\  y  ~~  suc  v
)  ->  E. w  e.  om  y  ~~  w
)
2118, 20sylan 471 . . . . . . . . . . . 12  |-  ( ( v  e.  om  /\  y  ~~  suc  v )  ->  E. w  e.  om  y  ~~  w )
22 isfi 7436 . . . . . . . . . . . 12  |-  ( y  e.  Fin  <->  E. w  e.  om  y  ~~  w
)
2321, 22sylibr 212 . . . . . . . . . . 11  |-  ( ( v  e.  om  /\  y  ~~  suc  v )  ->  y  e.  Fin )
24233adant2 1007 . . . . . . . . . 10  |-  ( ( v  e.  om  /\  A. x ( x  ~~  v  ->  ph )  /\  y  ~~  suc  v )  -> 
y  e.  Fin )
25 dif1en 7649 . . . . . . . . . . . . . . . 16  |-  ( ( v  e.  om  /\  y  ~~  suc  v  /\  z  e.  y )  ->  ( y  \  {
z } )  ~~  v )
26253expa 1188 . . . . . . . . . . . . . . 15  |-  ( ( ( v  e.  om  /\  y  ~~  suc  v
)  /\  z  e.  y )  ->  (
y  \  { z } )  ~~  v
)
27 vex 3074 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
28 difexg 4541 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  _V  ->  (
y  \  { z } )  e.  _V )
2927, 28ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( y 
\  { z } )  e.  _V
30 breq1 4396 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  \  { z } )  ->  ( x  ~~  v 
<->  ( y  \  {
z } )  ~~  v ) )
31 findcard.2 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  \  { z } )  ->  ( ph  <->  ch )
)
3230, 31imbi12d 320 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( y  \  { z } )  ->  ( ( x 
~~  v  ->  ph )  <->  ( ( y  \  {
z } )  ~~  v  ->  ch ) ) )
3329, 32spcv 3162 . . . . . . . . . . . . . . 15  |-  ( A. x ( x  ~~  v  ->  ph )  ->  (
( y  \  {
z } )  ~~  v  ->  ch ) )
3426, 33syl5com 30 . . . . . . . . . . . . . 14  |-  ( ( ( v  e.  om  /\  y  ~~  suc  v
)  /\  z  e.  y )  ->  ( A. x ( x  ~~  v  ->  ph )  ->  ch ) )
3534ralrimdva 2905 . . . . . . . . . . . . 13  |-  ( ( v  e.  om  /\  y  ~~  suc  v )  ->  ( A. x
( x  ~~  v  ->  ph )  ->  A. z  e.  y  ch )
)
3635imp 429 . . . . . . . . . . . 12  |-  ( ( ( v  e.  om  /\  y  ~~  suc  v
)  /\  A. x
( x  ~~  v  ->  ph ) )  ->  A. z  e.  y  ch )
3736an32s 802 . . . . . . . . . . 11  |-  ( ( ( v  e.  om  /\ 
A. x ( x 
~~  v  ->  ph )
)  /\  y  ~~  suc  v )  ->  A. z  e.  y  ch )
38373impa 1183 . . . . . . . . . 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 60 . . . . . . . . 9  |-  ( ( v  e.  om  /\  A. x ( x  ~~  v  ->  ph )  /\  y  ~~  suc  v )  ->  th )
41403exp 1187 . . . . . . . 8  |-  ( v  e.  om  ->  ( A. x ( x  ~~  v  ->  ph )  ->  (
y  ~~  suc  v  ->  th ) ) )
4241alrimdv 1688 . . . . . . 7  |-  ( v  e.  om  ->  ( A. x ( x  ~~  v  ->  ph )  ->  A. y
( y  ~~  suc  v  ->  th ) ) )
43 breq1 4396 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  ~~  suc  v  <->  y  ~~  suc  v ) )
44 findcard.3 . . . . . . . . 9  |-  ( x  =  y  ->  ( ph 
<->  th ) )
4543, 44imbi12d 320 . . . . . . . 8  |-  ( x  =  y  ->  (
( x  ~~  suc  v  ->  ph )  <->  ( y  ~~  suc  v  ->  th )
) )
4645cbvalv 1980 . . . . . . 7  |-  ( A. x ( x  ~~  suc  v  ->  ph )  <->  A. y ( y  ~~  suc  v  ->  th )
)
4742, 46syl6ibr 227 . . . . . 6  |-  ( v  e.  om  ->  ( A. x ( x  ~~  v  ->  ph )  ->  A. x
( x  ~~  suc  v  ->  ph ) ) )
485, 8, 11, 17, 47finds1 6608 . . . . 5  |-  ( w  e.  om  ->  A. x
( x  ~~  w  ->  ph ) )
494819.21bi 1806 . . . 4  |-  ( w  e.  om  ->  (
x  ~~  w  ->  ph ) )
5049rexlimiv 2934 . . 3  |-  ( E. w  e.  om  x  ~~  w  ->  ph )
512, 50sylbi 195 . 2  |-  ( x  e.  Fin  ->  ph )
521, 51vtoclga 3135 1  |-  ( A  e.  Fin  ->  ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1368    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   _Vcvv 3071    \ cdif 3426   (/)c0 3738   {csn 3978   class class class wbr 4393   suc csuc 4822   omcom 6579    ~~ cen 7410   Fincfn 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-om 6580  df-1o 7023  df-er 7204  df-en 7414  df-fin 7417
This theorem is referenced by:  xpfi  7687
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