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Theorem dif1en 7822
Description: If a set  A is equinumerous to the successor of a natural number  M, then  A with an element removed is equinumerous to  M. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)
Assertion
Ref Expression
dif1en  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( A  \  { X } )  ~~  M
)

Proof of Theorem dif1en
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 peano2 6732 . . . . 5  |-  ( M  e.  om  ->  suc  M  e.  om )
2 breq2 4399 . . . . . . 7  |-  ( x  =  suc  M  -> 
( A  ~~  x  <->  A 
~~  suc  M )
)
32rspcev 3136 . . . . . 6  |-  ( ( suc  M  e.  om  /\  A  ~~  suc  M
)  ->  E. x  e.  om  A  ~~  x
)
4 isfi 7611 . . . . . 6  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
53, 4sylibr 217 . . . . 5  |-  ( ( suc  M  e.  om  /\  A  ~~  suc  M
)  ->  A  e.  Fin )
61, 5sylan 479 . . . 4  |-  ( ( M  e.  om  /\  A  ~~  suc  M )  ->  A  e.  Fin )
7 diffi 7821 . . . . 5  |-  ( A  e.  Fin  ->  ( A  \  { X }
)  e.  Fin )
8 isfi 7611 . . . . 5  |-  ( ( A  \  { X } )  e.  Fin  <->  E. x  e.  om  ( A  \  { X }
)  ~~  x )
97, 8sylib 201 . . . 4  |-  ( A  e.  Fin  ->  E. x  e.  om  ( A  \  { X } )  ~~  x )
106, 9syl 17 . . 3  |-  ( ( M  e.  om  /\  A  ~~  suc  M )  ->  E. x  e.  om  ( A  \  { X } )  ~~  x
)
11103adant3 1050 . 2  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  E. x  e.  om  ( A  \  { X } )  ~~  x
)
12 vex 3034 . . . . . . . 8  |-  x  e. 
_V
13 en2sn 7667 . . . . . . . 8  |-  ( ( X  e.  A  /\  x  e.  _V )  ->  { X }  ~~  { x } )
1412, 13mpan2 685 . . . . . . 7  |-  ( X  e.  A  ->  { X }  ~~  { x }
)
15 nnord 6719 . . . . . . . 8  |-  ( x  e.  om  ->  Ord  x )
16 orddisj 5468 . . . . . . . 8  |-  ( Ord  x  ->  ( x  i^i  { x } )  =  (/) )
1715, 16syl 17 . . . . . . 7  |-  ( x  e.  om  ->  (
x  i^i  { x } )  =  (/) )
18 incom 3616 . . . . . . . . . 10  |-  ( ( A  \  { X } )  i^i  { X } )  =  ( { X }  i^i  ( A  \  { X } ) )
19 disjdif 3830 . . . . . . . . . 10  |-  ( { X }  i^i  ( A  \  { X }
) )  =  (/)
2018, 19eqtri 2493 . . . . . . . . 9  |-  ( ( A  \  { X } )  i^i  { X } )  =  (/)
21 unen 7670 . . . . . . . . . 10  |-  ( ( ( ( A  \  { X } )  ~~  x  /\  { X }  ~~  { x } )  /\  ( ( ( A  \  { X } )  i^i  { X } )  =  (/)  /\  ( x  i^i  {
x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2221an4s 842 . . . . . . . . 9  |-  ( ( ( ( A  \  { X } )  ~~  x  /\  ( ( A 
\  { X }
)  i^i  { X } )  =  (/) )  /\  ( { X }  ~~  { x }  /\  ( x  i^i  {
x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2320, 22mpanl2 695 . . . . . . . 8  |-  ( ( ( A  \  { X } )  ~~  x  /\  ( { X }  ~~  { x }  /\  ( x  i^i  { x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2423expcom 442 . . . . . . 7  |-  ( ( { X }  ~~  { x }  /\  (
x  i^i  { x } )  =  (/) )  ->  ( ( A 
\  { X }
)  ~~  x  ->  ( ( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) ) )
2514, 17, 24syl2an 485 . . . . . 6  |-  ( ( X  e.  A  /\  x  e.  om )  ->  ( ( A  \  { X } )  ~~  x  ->  ( ( A 
\  { X }
)  u.  { X } )  ~~  (
x  u.  { x } ) ) )
26253ad2antl3 1194 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( ( A  \  { X }
)  ~~  x  ->  ( ( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) ) )
27 difsnid 4109 . . . . . . . . 9  |-  ( X  e.  A  ->  (
( A  \  { X } )  u.  { X } )  =  A )
28 df-suc 5436 . . . . . . . . . . 11  |-  suc  x  =  ( x  u. 
{ x } )
2928eqcomi 2480 . . . . . . . . . 10  |-  ( x  u.  { x }
)  =  suc  x
3029a1i 11 . . . . . . . . 9  |-  ( X  e.  A  ->  (
x  u.  { x } )  =  suc  x )
3127, 30breq12d 4408 . . . . . . . 8  |-  ( X  e.  A  ->  (
( ( A  \  { X } )  u. 
{ X } ) 
~~  ( x  u. 
{ x } )  <-> 
A  ~~  suc  x ) )
32313ad2ant3 1053 . . . . . . 7  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( ( ( A 
\  { X }
)  u.  { X } )  ~~  (
x  u.  { x } )  <->  A  ~~  suc  x ) )
3332adantr 472 . . . . . 6  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } )  <->  A  ~~  suc  x ) )
34 ensym 7636 . . . . . . . . . . 11  |-  ( A 
~~  suc  M  ->  suc 
M  ~~  A )
35 entr 7639 . . . . . . . . . . . . 13  |-  ( ( suc  M  ~~  A  /\  A  ~~  suc  x
)  ->  suc  M  ~~  suc  x )
36 peano2 6732 . . . . . . . . . . . . . 14  |-  ( x  e.  om  ->  suc  x  e.  om )
37 nneneq 7773 . . . . . . . . . . . . . 14  |-  ( ( suc  M  e.  om  /\ 
suc  x  e.  om )  ->  ( suc  M  ~~  suc  x  <->  suc  M  =  suc  x ) )
3836, 37sylan2 482 . . . . . . . . . . . . 13  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( suc  M  ~~  suc  x  <->  suc  M  =  suc  x ) )
3935, 38syl5ib 227 . . . . . . . . . . . 12  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( ( suc  M  ~~  A  /\  A  ~~  suc  x )  ->  suc  M  =  suc  x ) )
4039expd 443 . . . . . . . . . . 11  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( suc  M  ~~  A  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) ) )
4134, 40syl5 32 . . . . . . . . . 10  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( A  ~~  suc  M  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) ) )
421, 41sylan 479 . . . . . . . . 9  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( A  ~~  suc  M  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) ) )
4342imp 436 . . . . . . . 8  |-  ( ( ( M  e.  om  /\  x  e.  om )  /\  A  ~~  suc  M
)  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
4443an32s 821 . . . . . . 7  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M
)  /\  x  e.  om )  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
45443adantl3 1188 . . . . . 6  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
4633, 45sylbid 223 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } )  ->  suc  M  =  suc  x ) )
47 peano4 6734 . . . . . . 7  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( suc  M  =  suc  x  <->  M  =  x ) )
4847biimpd 212 . . . . . 6  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( suc  M  =  suc  x  ->  M  =  x ) )
49483ad2antl1 1192 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( suc  M  =  suc  x  ->  M  =  x )
)
5026, 46, 493syld 56 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( ( A  \  { X }
)  ~~  x  ->  M  =  x ) )
51 breq2 4399 . . . . 5  |-  ( M  =  x  ->  (
( A  \  { X } )  ~~  M  <->  ( A  \  { X } )  ~~  x
) )
5251biimprcd 233 . . . 4  |-  ( ( A  \  { X } )  ~~  x  ->  ( M  =  x  ->  ( A  \  { X } )  ~~  M ) )
5350, 52sylcom 29 . . 3  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( ( A  \  { X }
)  ~~  x  ->  ( A  \  { X } )  ~~  M
) )
5453rexlimdva 2871 . 2  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( E. x  e. 
om  ( A  \  { X } )  ~~  x  ->  ( A  \  { X } )  ~~  M ) )
5511, 54mpd 15 1  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( A  \  { X } )  ~~  M
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   E.wrex 2757   _Vcvv 3031    \ cdif 3387    u. cun 3388    i^i cin 3389   (/)c0 3722   {csn 3959   class class class wbr 4395   Ord word 5429   suc csuc 5432   omcom 6711    ~~ cen 7584   Fincfn 7587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-1o 7200  df-er 7381  df-en 7588  df-fin 7591
This theorem is referenced by:  enp1i  7824  findcard  7828  findcard2  7829  en2eleq  8457  en2other2  8458  mreexexlem4d  15631  f1otrspeq  17166  pmtrf  17174  pmtrmvd  17175  pmtrfinv  17180
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