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Theorem dif1en 7755
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 6705 . . . . 5  |-  ( M  e.  om  ->  suc  M  e.  om )
2 breq2 4441 . . . . . . 7  |-  ( x  =  suc  M  -> 
( A  ~~  x  <->  A 
~~  suc  M )
)
32rspcev 3196 . . . . . 6  |-  ( ( suc  M  e.  om  /\  A  ~~  suc  M
)  ->  E. x  e.  om  A  ~~  x
)
4 isfi 7541 . . . . . 6  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
53, 4sylibr 212 . . . . 5  |-  ( ( suc  M  e.  om  /\  A  ~~  suc  M
)  ->  A  e.  Fin )
61, 5sylan 471 . . . 4  |-  ( ( M  e.  om  /\  A  ~~  suc  M )  ->  A  e.  Fin )
7 diffi 7753 . . . . 5  |-  ( A  e.  Fin  ->  ( A  \  { X }
)  e.  Fin )
8 isfi 7541 . . . . 5  |-  ( ( A  \  { X } )  e.  Fin  <->  E. x  e.  om  ( A  \  { X }
)  ~~  x )
97, 8sylib 196 . . . 4  |-  ( A  e.  Fin  ->  E. x  e.  om  ( A  \  { X } )  ~~  x )
106, 9syl 16 . . 3  |-  ( ( M  e.  om  /\  A  ~~  suc  M )  ->  E. x  e.  om  ( A  \  { X } )  ~~  x
)
11103adant3 1017 . 2  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  E. x  e.  om  ( A  \  { X } )  ~~  x
)
12 vex 3098 . . . . . . . 8  |-  x  e. 
_V
13 en2sn 7597 . . . . . . . 8  |-  ( ( X  e.  A  /\  x  e.  _V )  ->  { X }  ~~  { x } )
1412, 13mpan2 671 . . . . . . 7  |-  ( X  e.  A  ->  { X }  ~~  { x }
)
15 nnord 6693 . . . . . . . 8  |-  ( x  e.  om  ->  Ord  x )
16 orddisj 4906 . . . . . . . 8  |-  ( Ord  x  ->  ( x  i^i  { x } )  =  (/) )
1715, 16syl 16 . . . . . . 7  |-  ( x  e.  om  ->  (
x  i^i  { x } )  =  (/) )
18 incom 3676 . . . . . . . . . 10  |-  ( ( A  \  { X } )  i^i  { X } )  =  ( { X }  i^i  ( A  \  { X } ) )
19 disjdif 3886 . . . . . . . . . 10  |-  ( { X }  i^i  ( A  \  { X }
) )  =  (/)
2018, 19eqtri 2472 . . . . . . . . 9  |-  ( ( A  \  { X } )  i^i  { X } )  =  (/)
21 unen 7600 . . . . . . . . . 10  |-  ( ( ( ( A  \  { X } )  ~~  x  /\  { X }  ~~  { x } )  /\  ( ( ( A  \  { X } )  i^i  { X } )  =  (/)  /\  ( x  i^i  {
x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2221an4s 826 . . . . . . . . 9  |-  ( ( ( ( A  \  { X } )  ~~  x  /\  ( ( A 
\  { X }
)  i^i  { X } )  =  (/) )  /\  ( { X }  ~~  { x }  /\  ( x  i^i  {
x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2320, 22mpanl2 681 . . . . . . . 8  |-  ( ( ( A  \  { X } )  ~~  x  /\  ( { X }  ~~  { x }  /\  ( x  i^i  { x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2423expcom 435 . . . . . . 7  |-  ( ( { X }  ~~  { x }  /\  (
x  i^i  { x } )  =  (/) )  ->  ( ( A 
\  { X }
)  ~~  x  ->  ( ( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) ) )
2514, 17, 24syl2an 477 . . . . . 6  |-  ( ( X  e.  A  /\  x  e.  om )  ->  ( ( A  \  { X } )  ~~  x  ->  ( ( A 
\  { X }
)  u.  { X } )  ~~  (
x  u.  { x } ) ) )
26253ad2antl3 1161 . . . . 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 4161 . . . . . . . . 9  |-  ( X  e.  A  ->  (
( A  \  { X } )  u.  { X } )  =  A )
28 df-suc 4874 . . . . . . . . . . 11  |-  suc  x  =  ( x  u. 
{ x } )
2928eqcomi 2456 . . . . . . . . . 10  |-  ( x  u.  { x }
)  =  suc  x
3029a1i 11 . . . . . . . . 9  |-  ( X  e.  A  ->  (
x  u.  { x } )  =  suc  x )
3127, 30breq12d 4450 . . . . . . . 8  |-  ( X  e.  A  ->  (
( ( A  \  { X } )  u. 
{ X } ) 
~~  ( x  u. 
{ x } )  <-> 
A  ~~  suc  x ) )
32313ad2ant3 1020 . . . . . . 7  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( ( ( A 
\  { X }
)  u.  { X } )  ~~  (
x  u.  { x } )  <->  A  ~~  suc  x ) )
3332adantr 465 . . . . . 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 7566 . . . . . . . . . . 11  |-  ( A 
~~  suc  M  ->  suc 
M  ~~  A )
35 entr 7569 . . . . . . . . . . . . 13  |-  ( ( suc  M  ~~  A  /\  A  ~~  suc  x
)  ->  suc  M  ~~  suc  x )
36 peano2 6705 . . . . . . . . . . . . . 14  |-  ( x  e.  om  ->  suc  x  e.  om )
37 nneneq 7702 . . . . . . . . . . . . . 14  |-  ( ( suc  M  e.  om  /\ 
suc  x  e.  om )  ->  ( suc  M  ~~  suc  x  <->  suc  M  =  suc  x ) )
3836, 37sylan2 474 . . . . . . . . . . . . 13  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( suc  M  ~~  suc  x  <->  suc  M  =  suc  x ) )
3935, 38syl5ib 219 . . . . . . . . . . . 12  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( ( suc  M  ~~  A  /\  A  ~~  suc  x )  ->  suc  M  =  suc  x ) )
4039expd 436 . . . . . . . . . . 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 471 . . . . . . . . 9  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( A  ~~  suc  M  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) ) )
4342imp 429 . . . . . . . 8  |-  ( ( ( M  e.  om  /\  x  e.  om )  /\  A  ~~  suc  M
)  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
4443an32s 804 . . . . . . 7  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M
)  /\  x  e.  om )  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
45443adantl3 1155 . . . . . 6  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
4633, 45sylbid 215 . . . . 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 6707 . . . . . . 7  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( suc  M  =  suc  x  <->  M  =  x ) )
4847biimpd 207 . . . . . 6  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( suc  M  =  suc  x  ->  M  =  x ) )
49483ad2antl1 1159 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( suc  M  =  suc  x  ->  M  =  x )
)
5026, 46, 493syld 55 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( ( A  \  { X }
)  ~~  x  ->  M  =  x ) )
51 breq2 4441 . . . . 5  |-  ( M  =  x  ->  (
( A  \  { X } )  ~~  M  <->  ( A  \  { X } )  ~~  x
) )
5251biimprcd 225 . . . 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 2935 . 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 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   E.wrex 2794   _Vcvv 3095    \ cdif 3458    u. cun 3459    i^i cin 3460   (/)c0 3770   {csn 4014   class class class wbr 4437   Ord word 4867   suc csuc 4870   omcom 6685    ~~ cen 7515   Fincfn 7518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-1o 7132  df-er 7313  df-en 7519  df-fin 7522
This theorem is referenced by:  enp1i  7757  findcard  7761  findcard2  7762  en2eleq  8389  en2other2  8390  mreexexlem4d  15025  f1otrspeq  16450  pmtrf  16458  pmtrmvd  16459  pmtrfinv  16464
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