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Theorem dif1card 8389
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 7752 . . 3  |-  ( A  e.  Fin  ->  ( A  \  { X }
)  e.  Fin )
2 isfi 7540 . . . 4  |-  ( ( A  \  { X } )  e.  Fin  <->  E. m  e.  om  ( A  \  { X }
)  ~~  m )
3 simp3 998 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( A  \  { X } )  ~~  m
)
4 en2sn 7596 . . . . . . . . . . . 12  |-  ( ( X  e.  A  /\  m  e.  om )  ->  { X }  ~~  { m } )
543adant3 1016 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  { X }  ~~  { m } )
6 incom 3691 . . . . . . . . . . . . 13  |-  ( ( A  \  { X } )  i^i  { X } )  =  ( { X }  i^i  ( A  \  { X } ) )
7 disjdif 3899 . . . . . . . . . . . . 13  |-  ( { X }  i^i  ( A  \  { X }
) )  =  (/)
86, 7eqtri 2496 . . . . . . . . . . . 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 6693 . . . . . . . . . . . . . 14  |-  ( m  e.  om  ->  Ord  m )
11 ordirr 4896 . . . . . . . . . . . . . 14  |-  ( Ord  m  ->  -.  m  e.  m )
1210, 11syl 16 . . . . . . . . . . . . 13  |-  ( m  e.  om  ->  -.  m  e.  m )
13 disjsn 4088 . . . . . . . . . . . . 13  |-  ( ( m  i^i  { m } )  =  (/)  <->  -.  m  e.  m )
1412, 13sylibr 212 . . . . . . . . . . . 12  |-  ( m  e.  om  ->  (
m  i^i  { m } )  =  (/) )
15143ad2ant2 1018 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( m  i^i  {
m } )  =  (/) )
16 unen 7599 . . . . . . . . . . 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 1229 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( ( A  \  { X } )  u. 
{ X } ) 
~~  ( m  u. 
{ m } ) )
18 difsnid 4173 . . . . . . . . . . . 12  |-  ( X  e.  A  ->  (
( A  \  { X } )  u.  { X } )  =  A )
19 df-suc 4884 . . . . . . . . . . . . . 14  |-  suc  m  =  ( m  u. 
{ m } )
2019eqcomi 2480 . . . . . . . . . . . . 13  |-  ( m  u.  { m }
)  =  suc  m
2120a1i 11 . . . . . . . . . . . 12  |-  ( X  e.  A  ->  (
m  u.  { m } )  =  suc  m )
2218, 21breq12d 4460 . . . . . . . . . . 11  |-  ( X  e.  A  ->  (
( ( A  \  { X } )  u. 
{ X } ) 
~~  ( m  u. 
{ m } )  <-> 
A  ~~  suc  m ) )
23223ad2ant1 1017 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( ( ( A 
\  { X }
)  u.  { X } )  ~~  (
m  u.  { m } )  <->  A  ~~  suc  m ) )
2417, 23mpbid 210 . . . . . . . . 9  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  A  ~~  suc  m
)
25 peano2 6705 . . . . . . . . . 10  |-  ( m  e.  om  ->  suc  m  e.  om )
26253ad2ant2 1018 . . . . . . . . 9  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  suc  m  e.  om )
27 cardennn 8365 . . . . . . . . 9  |-  ( ( A  ~~  suc  m  /\  suc  m  e.  om )  ->  ( card `  A
)  =  suc  m
)
2824, 26, 27syl2anc 661 . . . . . . . 8  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( card `  A
)  =  suc  m
)
29 cardennn 8365 . . . . . . . . . . 11  |-  ( ( ( A  \  { X } )  ~~  m  /\  m  e.  om )  ->  ( card `  ( A  \  { X }
) )  =  m )
3029ancoms 453 . . . . . . . . . 10  |-  ( ( m  e.  om  /\  ( A  \  { X } )  ~~  m
)  ->  ( card `  ( A  \  { X } ) )  =  m )
31303adant1 1014 . . . . . . . . 9  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( card `  ( A  \  { X }
) )  =  m )
32 suceq 4943 . . . . . . . . 9  |-  ( (
card `  ( A  \  { X } ) )  =  m  ->  suc  ( card `  ( A  \  { X }
) )  =  suc  m )
3331, 32syl 16 . . . . . . . 8  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  suc  ( card `  ( A  \  { X }
) )  =  suc  m )
3428, 33eqtr4d 2511 . . . . . . 7  |-  ( ( X  e.  A  /\  m  e.  om  /\  ( A  \  { X }
)  ~~  m )  ->  ( card `  A
)  =  suc  ( card `  ( A  \  { X } ) ) )
35343expib 1199 . . . . . 6  |-  ( X  e.  A  ->  (
( m  e.  om  /\  ( A  \  { X } )  ~~  m
)  ->  ( card `  A )  =  suc  ( card `  ( A  \  { X } ) ) ) )
3635com12 31 . . . . 5  |-  ( ( m  e.  om  /\  ( A  \  { X } )  ~~  m
)  ->  ( X  e.  A  ->  ( card `  A )  =  suc  ( card `  ( A  \  { X } ) ) ) )
3736rexlimiva 2951 . . . 4  |-  ( E. m  e.  om  ( A  \  { X }
)  ~~  m  ->  ( X  e.  A  -> 
( card `  A )  =  suc  ( card `  ( A  \  { X }
) ) ) )
382, 37sylbi 195 . . 3  |-  ( ( A  \  { X } )  e.  Fin  ->  ( X  e.  A  ->  ( card `  A
)  =  suc  ( card `  ( A  \  { X } ) ) ) )
391, 38syl 16 . 2  |-  ( A  e.  Fin  ->  ( X  e.  A  ->  (
card `  A )  =  suc  ( card `  ( A  \  { X }
) ) ) )
4039imp 429 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815    \ cdif 3473    u. cun 3474    i^i cin 3475   (/)c0 3785   {csn 4027   class class class wbr 4447   Ord word 4877   suc csuc 4880   ` cfv 5588   omcom 6685    ~~ cen 7514   Fincfn 7517   cardccrd 8317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6686  df-1o 7131  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321
This theorem is referenced by: (None)
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