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Theorem dif1card 8440
Description: The cardinality of a nonempty finite set is one greater than the cardinality of the set with one element removed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
dif1card  |-  ( ( A  e.  Fin  /\  X  e.  A )  ->  ( card `  A
)  =  suc  ( card `  ( A  \  { X } ) ) )

Proof of Theorem dif1card
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 diffi 7809 . . 3  |-  ( A  e.  Fin  ->  ( A  \  { X }
)  e.  Fin )
2 isfi 7600 . . . 4  |-  ( ( A  \  { X } )  e.  Fin  <->  E. m  e.  om  ( A  \  { X }
)  ~~  m )
3 simp3 1007 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( A  \  { X } )  ~~  m
)
4 en2sn 7656 . . . . . . . . . . . 12  |-  ( ( X  e.  A  /\  m  e.  om )  ->  { X }  ~~  { m } )
543adant3 1025 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  { X }  ~~  { m } )
6 incom 3661 . . . . . . . . . . . . 13  |-  ( ( A  \  { X } )  i^i  { X } )  =  ( { X }  i^i  ( A  \  { X } ) )
7 disjdif 3873 . . . . . . . . . . . . 13  |-  ( { X }  i^i  ( A  \  { X }
) )  =  (/)
86, 7eqtri 2458 . . . . . . . . . . . 12  |-  ( ( A  \  { X } )  i^i  { X } )  =  (/)
98a1i 11 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( ( A  \  { X } )  i^i 
{ X } )  =  (/) )
10 nnord 6714 . . . . . . . . . . . . . 14  |-  ( m  e.  om  ->  Ord  m )
11 ordirr 5460 . . . . . . . . . . . . . 14  |-  ( Ord  m  ->  -.  m  e.  m )
1210, 11syl 17 . . . . . . . . . . . . 13  |-  ( m  e.  om  ->  -.  m  e.  m )
13 disjsn 4063 . . . . . . . . . . . . 13  |-  ( ( m  i^i  { m } )  =  (/)  <->  -.  m  e.  m )
1412, 13sylibr 215 . . . . . . . . . . . 12  |-  ( m  e.  om  ->  (
m  i^i  { m } )  =  (/) )
15143ad2ant2 1027 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( m  i^i  {
m } )  =  (/) )
16 unen 7659 . . . . . . . . . . 11  |-  ( ( ( ( A  \  { X } )  ~~  m  /\  { X }  ~~  { m } )  /\  ( ( ( A  \  { X } )  i^i  { X } )  =  (/)  /\  ( m  i^i  {
m } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
m  u.  { m } ) )
173, 5, 9, 15, 16syl22anc 1265 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( ( A  \  { X } )  u. 
{ X } ) 
~~  ( m  u. 
{ m } ) )
18 difsnid 4149 . . . . . . . . . . . 12  |-  ( X  e.  A  ->  (
( A  \  { X } )  u.  { X } )  =  A )
19 df-suc 5448 . . . . . . . . . . . . . 14  |-  suc  m  =  ( m  u. 
{ m } )
2019eqcomi 2442 . . . . . . . . . . . . 13  |-  ( m  u.  { m }
)  =  suc  m
2120a1i 11 . . . . . . . . . . . 12  |-  ( X  e.  A  ->  (
m  u.  { m } )  =  suc  m )
2218, 21breq12d 4439 . . . . . . . . . . 11  |-  ( X  e.  A  ->  (
( ( A  \  { X } )  u. 
{ X } ) 
~~  ( m  u. 
{ m } )  <-> 
A  ~~  suc  m ) )
23223ad2ant1 1026 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( ( ( A 
\  { X }
)  u.  { X } )  ~~  (
m  u.  { m } )  <->  A  ~~  suc  m ) )
2417, 23mpbid 213 . . . . . . . . 9  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  A  ~~  suc  m
)
25 peano2 6727 . . . . . . . . . 10  |-  ( m  e.  om  ->  suc  m  e.  om )
26253ad2ant2 1027 . . . . . . . . 9  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  suc  m  e.  om )
27 cardennn 8416 . . . . . . . . 9  |-  ( ( A  ~~  suc  m  /\  suc  m  e.  om )  ->  ( card `  A
)  =  suc  m
)
2824, 26, 27syl2anc 665 . . . . . . . 8  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( card `  A
)  =  suc  m
)
29 cardennn 8416 . . . . . . . . . . 11  |-  ( ( ( A  \  { X } )  ~~  m  /\  m  e.  om )  ->  ( card `  ( A  \  { X }
) )  =  m )
3029ancoms 454 . . . . . . . . . 10  |-  ( ( m  e.  om  /\  ( A  \  { X } )  ~~  m
)  ->  ( card `  ( A  \  { X } ) )  =  m )
31303adant1 1023 . . . . . . . . 9  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( card `  ( A  \  { X }
) )  =  m )
32 suceq 5507 . . . . . . . . 9  |-  ( (
card `  ( A  \  { X } ) )  =  m  ->  suc  ( card `  ( A  \  { X }
) )  =  suc  m )
3331, 32syl 17 . . . . . . . 8  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  suc  ( card `  ( A  \  { X }
) )  =  suc  m )
3428, 33eqtr4d 2473 . . . . . . 7  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( card `  A
)  =  suc  ( card `  ( A  \  { X } ) ) )
35343expib 1208 . . . . . 6  |-  ( X  e.  A  ->  (
( m  e.  om  /\  ( A  \  { X } )  ~~  m
)  ->  ( card `  A )  =  suc  ( card `  ( A  \  { X } ) ) ) )
3635com12 32 . . . . 5  |-  ( ( m  e.  om  /\  ( A  \  { X } )  ~~  m
)  ->  ( X  e.  A  ->  ( card `  A )  =  suc  ( card `  ( A  \  { X } ) ) ) )
3736rexlimiva 2920 . . . 4  |-  ( E. m  e.  om  ( A  \  { X }
)  ~~  m  ->  ( X  e.  A  -> 
( card `  A )  =  suc  ( card `  ( A  \  { X }
) ) ) )
382, 37sylbi 198 . . 3  |-  ( ( A  \  { X } )  e.  Fin  ->  ( X  e.  A  ->  ( card `  A
)  =  suc  ( card `  ( A  \  { X } ) ) ) )
391, 38syl 17 . 2  |-  ( A  e.  Fin  ->  ( X  e.  A  ->  (
card `  A )  =  suc  ( card `  ( A  \  { X }
) ) ) )
4039imp 430 1  |-  ( ( A  e.  Fin  /\  X  e.  A )  ->  ( card `  A
)  =  suc  ( card `  ( A  \  { X } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   E.wrex 2783    \ cdif 3439    u. cun 3440    i^i cin 3441   (/)c0 3767   {csn 4002   class class class wbr 4426   Ord word 5441   suc csuc 5444   ` cfv 5601   omcom 6706    ~~ cen 7574   Fincfn 7577   cardccrd 8368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-1o 7190  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372
This theorem is referenced by: (None)
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