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Theorem dif1en 7333
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 4857 . . . . 5  |-  ( M  e.  om  ->  suc  M  e.  om )
2 breq2 4208 . . . . . . 7  |-  ( x  =  suc  M  -> 
( A  ~~  x  <->  A 
~~  suc  M )
)
32rspcev 3044 . . . . . 6  |-  ( ( suc  M  e.  om  /\  A  ~~  suc  M
)  ->  E. x  e.  om  A  ~~  x
)
4 isfi 7123 . . . . . 6  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
53, 4sylibr 204 . . . . 5  |-  ( ( suc  M  e.  om  /\  A  ~~  suc  M
)  ->  A  e.  Fin )
61, 5sylan 458 . . . 4  |-  ( ( M  e.  om  /\  A  ~~  suc  M )  ->  A  e.  Fin )
7 diffi 7331 . . . . 5  |-  ( A  e.  Fin  ->  ( A  \  { X }
)  e.  Fin )
8 isfi 7123 . . . . 5  |-  ( ( A  \  { X } )  e.  Fin  <->  E. x  e.  om  ( A  \  { X }
)  ~~  x )
97, 8sylib 189 . . . 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 977 . 2  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  E. x  e.  om  ( A  \  { X } )  ~~  x
)
12 vex 2951 . . . . . . . 8  |-  x  e. 
_V
13 en2sn 7178 . . . . . . . 8  |-  ( ( X  e.  A  /\  x  e.  _V )  ->  { X }  ~~  { x } )
1412, 13mpan2 653 . . . . . . 7  |-  ( X  e.  A  ->  { X }  ~~  { x }
)
15 nnord 4845 . . . . . . . 8  |-  ( x  e.  om  ->  Ord  x )
16 orddisj 4611 . . . . . . . 8  |-  ( Ord  x  ->  ( x  i^i  { x } )  =  (/) )
1715, 16syl 16 . . . . . . 7  |-  ( x  e.  om  ->  (
x  i^i  { x } )  =  (/) )
18 incom 3525 . . . . . . . . . 10  |-  ( ( A  \  { X } )  i^i  { X } )  =  ( { X }  i^i  ( A  \  { X } ) )
19 disjdif 3692 . . . . . . . . . 10  |-  ( { X }  i^i  ( A  \  { X }
) )  =  (/)
2018, 19eqtri 2455 . . . . . . . . 9  |-  ( ( A  \  { X } )  i^i  { X } )  =  (/)
21 unen 7181 . . . . . . . . . 10  |-  ( ( ( ( A  \  { X } )  ~~  x  /\  { X }  ~~  { x } )  /\  ( ( ( A  \  { X } )  i^i  { X } )  =  (/)  /\  ( x  i^i  {
x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2221an4s 800 . . . . . . . . 9  |-  ( ( ( ( A  \  { X } )  ~~  x  /\  ( ( A 
\  { X }
)  i^i  { X } )  =  (/) )  /\  ( { X }  ~~  { x }  /\  ( x  i^i  {
x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2320, 22mpanl2 663 . . . . . . . 8  |-  ( ( ( A  \  { X } )  ~~  x  /\  ( { X }  ~~  { x }  /\  ( x  i^i  { x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2423expcom 425 . . . . . . 7  |-  ( ( { X }  ~~  { x }  /\  (
x  i^i  { x } )  =  (/) )  ->  ( ( A 
\  { X }
)  ~~  x  ->  ( ( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) ) )
2514, 17, 24syl2an 464 . . . . . 6  |-  ( ( X  e.  A  /\  x  e.  om )  ->  ( ( A  \  { X } )  ~~  x  ->  ( ( A 
\  { X }
)  u.  { X } )  ~~  (
x  u.  { x } ) ) )
26253ad2antl3 1121 . . . . 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 3936 . . . . . . . . 9  |-  ( X  e.  A  ->  (
( A  \  { X } )  u.  { X } )  =  A )
28 df-suc 4579 . . . . . . . . . . 11  |-  suc  x  =  ( x  u. 
{ x } )
2928eqcomi 2439 . . . . . . . . . 10  |-  ( x  u.  { x }
)  =  suc  x
3029a1i 11 . . . . . . . . 9  |-  ( X  e.  A  ->  (
x  u.  { x } )  =  suc  x )
3127, 30breq12d 4217 . . . . . . . 8  |-  ( X  e.  A  ->  (
( ( A  \  { X } )  u. 
{ X } ) 
~~  ( x  u. 
{ x } )  <-> 
A  ~~  suc  x ) )
32313ad2ant3 980 . . . . . . 7  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( ( ( A 
\  { X }
)  u.  { X } )  ~~  (
x  u.  { x } )  <->  A  ~~  suc  x ) )
3332adantr 452 . . . . . 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 7148 . . . . . . . . . . 11  |-  ( A 
~~  suc  M  ->  suc 
M  ~~  A )
35 entr 7151 . . . . . . . . . . . . 13  |-  ( ( suc  M  ~~  A  /\  A  ~~  suc  x
)  ->  suc  M  ~~  suc  x )
36 peano2 4857 . . . . . . . . . . . . . 14  |-  ( x  e.  om  ->  suc  x  e.  om )
37 nneneq 7282 . . . . . . . . . . . . . 14  |-  ( ( suc  M  e.  om  /\ 
suc  x  e.  om )  ->  ( suc  M  ~~  suc  x  <->  suc  M  =  suc  x ) )
3836, 37sylan2 461 . . . . . . . . . . . . 13  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( suc  M  ~~  suc  x  <->  suc  M  =  suc  x ) )
3935, 38syl5ib 211 . . . . . . . . . . . 12  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( ( suc  M  ~~  A  /\  A  ~~  suc  x )  ->  suc  M  =  suc  x ) )
4039exp3a 426 . . . . . . . . . . 11  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( suc  M  ~~  A  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) ) )
4134, 40syl5 30 . . . . . . . . . 10  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( A  ~~  suc  M  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) ) )
421, 41sylan 458 . . . . . . . . 9  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( A  ~~  suc  M  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) ) )
4342imp 419 . . . . . . . 8  |-  ( ( ( M  e.  om  /\  x  e.  om )  /\  A  ~~  suc  M
)  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
4443an32s 780 . . . . . . 7  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M
)  /\  x  e.  om )  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
45443adantl3 1115 . . . . . 6  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
4633, 45sylbid 207 . . . . 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 4859 . . . . . . 7  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( suc  M  =  suc  x  <->  M  =  x ) )
4847biimpd 199 . . . . . 6  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( suc  M  =  suc  x  ->  M  =  x ) )
49483ad2antl1 1119 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( suc  M  =  suc  x  ->  M  =  x )
)
5026, 46, 493syld 53 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( ( A  \  { X }
)  ~~  x  ->  M  =  x ) )
51 breq2 4208 . . . . 5  |-  ( M  =  x  ->  (
( A  \  { X } )  ~~  M  <->  ( A  \  { X } )  ~~  x
) )
5251biimprcd 217 . . . 4  |-  ( ( A  \  { X } )  ~~  x  ->  ( M  =  x  ->  ( A  \  { X } )  ~~  M ) )
5350, 52sylcom 27 . . 3  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( ( A  \  { X }
)  ~~  x  ->  ( A  \  { X } )  ~~  M
) )
5453rexlimdva 2822 . 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311   (/)c0 3620   {csn 3806   class class class wbr 4204   Ord word 4572   suc csuc 4575   omcom 4837    ~~ cen 7098   Fincfn 7101
This theorem is referenced by:  enp1i  7335  findcard  7339  findcard2  7340  mreexexlem4d  13864  en2eleq  27349  en2other2  27350  f1otrspeq  27358  pmtrf  27365  pmtrmvd  27366  pmtrfinv  27370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1o 6716  df-er 6897  df-en 7102  df-fin 7105
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