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Theorem dif1en 7107
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 4692 . . . . 5  |-  ( M  e.  om  ->  suc  M  e.  om )
2 breq2 4043 . . . . . . 7  |-  ( x  =  suc  M  -> 
( A  ~~  x  <->  A 
~~  suc  M )
)
32rspcev 2897 . . . . . 6  |-  ( ( suc  M  e.  om  /\  A  ~~  suc  M
)  ->  E. x  e.  om  A  ~~  x
)
4 isfi 6901 . . . . . 6  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
53, 4sylibr 203 . . . . 5  |-  ( ( suc  M  e.  om  /\  A  ~~  suc  M
)  ->  A  e.  Fin )
61, 5sylan 457 . . . 4  |-  ( ( M  e.  om  /\  A  ~~  suc  M )  ->  A  e.  Fin )
7 diffi 7105 . . . . 5  |-  ( A  e.  Fin  ->  ( A  \  { X }
)  e.  Fin )
8 isfi 6901 . . . . 5  |-  ( ( A  \  { X } )  e.  Fin  <->  E. x  e.  om  ( A  \  { X }
)  ~~  x )
97, 8sylib 188 . . . 4  |-  ( A  e.  Fin  ->  E. x  e.  om  ( A  \  { X } )  ~~  x )
106, 9syl 15 . . 3  |-  ( ( M  e.  om  /\  A  ~~  suc  M )  ->  E. x  e.  om  ( A  \  { X } )  ~~  x
)
11103adant3 975 . 2  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  E. x  e.  om  ( A  \  { X } )  ~~  x
)
12 vex 2804 . . . . . . . 8  |-  x  e. 
_V
13 en2sn 6956 . . . . . . . 8  |-  ( ( X  e.  A  /\  x  e.  _V )  ->  { X }  ~~  { x } )
1412, 13mpan2 652 . . . . . . 7  |-  ( X  e.  A  ->  { X }  ~~  { x }
)
15 nnord 4680 . . . . . . . 8  |-  ( x  e.  om  ->  Ord  x )
16 orddisj 4446 . . . . . . . 8  |-  ( Ord  x  ->  ( x  i^i  { x } )  =  (/) )
1715, 16syl 15 . . . . . . 7  |-  ( x  e.  om  ->  (
x  i^i  { x } )  =  (/) )
18 incom 3374 . . . . . . . . . 10  |-  ( ( A  \  { X } )  i^i  { X } )  =  ( { X }  i^i  ( A  \  { X } ) )
19 disjdif 3539 . . . . . . . . . 10  |-  ( { X }  i^i  ( A  \  { X }
) )  =  (/)
2018, 19eqtri 2316 . . . . . . . . 9  |-  ( ( A  \  { X } )  i^i  { X } )  =  (/)
21 unen 6959 . . . . . . . . . 10  |-  ( ( ( ( A  \  { X } )  ~~  x  /\  { X }  ~~  { x } )  /\  ( ( ( A  \  { X } )  i^i  { X } )  =  (/)  /\  ( x  i^i  {
x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2221an4s 799 . . . . . . . . 9  |-  ( ( ( ( A  \  { X } )  ~~  x  /\  ( ( A 
\  { X }
)  i^i  { X } )  =  (/) )  /\  ( { X }  ~~  { x }  /\  ( x  i^i  {
x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2320, 22mpanl2 662 . . . . . . . 8  |-  ( ( ( A  \  { X } )  ~~  x  /\  ( { X }  ~~  { x }  /\  ( x  i^i  { x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2423expcom 424 . . . . . . 7  |-  ( ( { X }  ~~  { x }  /\  (
x  i^i  { x } )  =  (/) )  ->  ( ( A 
\  { X }
)  ~~  x  ->  ( ( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) ) )
2514, 17, 24syl2an 463 . . . . . 6  |-  ( ( X  e.  A  /\  x  e.  om )  ->  ( ( A  \  { X } )  ~~  x  ->  ( ( A 
\  { X }
)  u.  { X } )  ~~  (
x  u.  { x } ) ) )
26253ad2antl3 1119 . . . . 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 3777 . . . . . . . . 9  |-  ( X  e.  A  ->  (
( A  \  { X } )  u.  { X } )  =  A )
28 df-suc 4414 . . . . . . . . . . 11  |-  suc  x  =  ( x  u. 
{ x } )
2928eqcomi 2300 . . . . . . . . . 10  |-  ( x  u.  { x }
)  =  suc  x
3029a1i 10 . . . . . . . . 9  |-  ( X  e.  A  ->  (
x  u.  { x } )  =  suc  x )
3127, 30breq12d 4052 . . . . . . . 8  |-  ( X  e.  A  ->  (
( ( A  \  { X } )  u. 
{ X } ) 
~~  ( x  u. 
{ x } )  <-> 
A  ~~  suc  x ) )
32313ad2ant3 978 . . . . . . 7  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( ( ( A 
\  { X }
)  u.  { X } )  ~~  (
x  u.  { x } )  <->  A  ~~  suc  x ) )
3332adantr 451 . . . . . 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 6926 . . . . . . . . . . 11  |-  ( A 
~~  suc  M  ->  suc 
M  ~~  A )
35 entr 6929 . . . . . . . . . . . . 13  |-  ( ( suc  M  ~~  A  /\  A  ~~  suc  x
)  ->  suc  M  ~~  suc  x )
36 peano2 4692 . . . . . . . . . . . . . 14  |-  ( x  e.  om  ->  suc  x  e.  om )
37 nneneq 7060 . . . . . . . . . . . . . 14  |-  ( ( suc  M  e.  om  /\ 
suc  x  e.  om )  ->  ( suc  M  ~~  suc  x  <->  suc  M  =  suc  x ) )
3836, 37sylan2 460 . . . . . . . . . . . . 13  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( suc  M  ~~  suc  x  <->  suc  M  =  suc  x ) )
3935, 38syl5ib 210 . . . . . . . . . . . 12  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( ( suc  M  ~~  A  /\  A  ~~  suc  x )  ->  suc  M  =  suc  x ) )
4039exp3a 425 . . . . . . . . . . 11  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( suc  M  ~~  A  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) ) )
4134, 40syl5 28 . . . . . . . . . 10  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( A  ~~  suc  M  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) ) )
421, 41sylan 457 . . . . . . . . 9  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( A  ~~  suc  M  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) ) )
4342imp 418 . . . . . . . 8  |-  ( ( ( M  e.  om  /\  x  e.  om )  /\  A  ~~  suc  M
)  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
4443an32s 779 . . . . . . 7  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M
)  /\  x  e.  om )  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
45443adantl3 1113 . . . . . 6  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
4633, 45sylbid 206 . . . . 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 4694 . . . . . . 7  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( suc  M  =  suc  x  <->  M  =  x ) )
4847biimpd 198 . . . . . 6  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( suc  M  =  suc  x  ->  M  =  x ) )
49483ad2antl1 1117 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( suc  M  =  suc  x  ->  M  =  x )
)
5026, 46, 493syld 51 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( ( A  \  { X }
)  ~~  x  ->  M  =  x ) )
51 breq2 4043 . . . . 5  |-  ( M  =  x  ->  (
( A  \  { X } )  ~~  M  <->  ( A  \  { X } )  ~~  x
) )
5251biimprcd 216 . . . 4  |-  ( ( A  \  { X } )  ~~  x  ->  ( M  =  x  ->  ( A  \  { X } )  ~~  M ) )
5350, 52sylcom 25 . . 3  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( ( A  \  { X }
)  ~~  x  ->  ( A  \  { X } )  ~~  M
) )
5453rexlimdva 2680 . 2  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( E. x  e. 
om  ( A  \  { X } )  ~~  x  ->  ( A  \  { X } )  ~~  M ) )
5511, 54mpd 14 1  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( A  \  { X } )  ~~  M
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   class class class wbr 4039   Ord word 4407   suc csuc 4410   omcom 4672    ~~ cen 6876   Fincfn 6879
This theorem is referenced by:  enp1i  7109  findcard  7113  findcard2  7114  mreexexlem4d  13565  en2eleq  27484  en2other2  27485  f1otrspeq  27493  pmtrf  27500  pmtrmvd  27501  pmtrfinv  27505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-er 6676  df-en 6880  df-fin 6883
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