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Theorem dif1enOLD 7342
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.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
dif1en.1  |-  A  e. 
_V
Assertion
Ref Expression
dif1enOLD  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( A  \  { X } )  ~~  M
)

Proof of Theorem dif1enOLD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 peano2 4867 . . . . 5  |-  ( M  e.  om  ->  suc  M  e.  om )
2 breq2 4218 . . . . . . 7  |-  ( x  =  suc  M  -> 
( A  ~~  x  <->  A 
~~  suc  M )
)
32rspcev 3054 . . . . . 6  |-  ( ( suc  M  e.  om  /\  A  ~~  suc  M
)  ->  E. x  e.  om  A  ~~  x
)
4 isfi 7133 . . . . . 6  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
53, 4sylibr 205 . . . . 5  |-  ( ( suc  M  e.  om  /\  A  ~~  suc  M
)  ->  A  e.  Fin )
61, 5sylan 459 . . . 4  |-  ( ( M  e.  om  /\  A  ~~  suc  M )  ->  A  e.  Fin )
7 diffi 7341 . . . . 5  |-  ( A  e.  Fin  ->  ( A  \  { X }
)  e.  Fin )
8 isfi 7133 . . . . 5  |-  ( ( A  \  { X } )  e.  Fin  <->  E. x  e.  om  ( A  \  { X }
)  ~~  x )
97, 8sylib 190 . . . 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 978 . 2  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  E. x  e.  om  ( A  \  { X } )  ~~  x
)
12 vex 2961 . . . . . . . 8  |-  x  e. 
_V
13 en2sn 7188 . . . . . . . 8  |-  ( ( X  e.  A  /\  x  e.  _V )  ->  { X }  ~~  { x } )
1412, 13mpan2 654 . . . . . . 7  |-  ( X  e.  A  ->  { X }  ~~  { x }
)
15 nnord 4855 . . . . . . . 8  |-  ( x  e.  om  ->  Ord  x )
16 orddisj 4621 . . . . . . . 8  |-  ( Ord  x  ->  ( x  i^i  { x } )  =  (/) )
1715, 16syl 16 . . . . . . 7  |-  ( x  e.  om  ->  (
x  i^i  { x } )  =  (/) )
18 incom 3535 . . . . . . . . . 10  |-  ( ( A  \  { X } )  i^i  { X } )  =  ( { X }  i^i  ( A  \  { X } ) )
19 disjdif 3702 . . . . . . . . . 10  |-  ( { X }  i^i  ( A  \  { X }
) )  =  (/)
2018, 19eqtri 2458 . . . . . . . . 9  |-  ( ( A  \  { X } )  i^i  { X } )  =  (/)
21 unen 7191 . . . . . . . . . 10  |-  ( ( ( ( A  \  { X } )  ~~  x  /\  { X }  ~~  { x } )  /\  ( ( ( A  \  { X } )  i^i  { X } )  =  (/)  /\  ( x  i^i  {
x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2221an4s 801 . . . . . . . . 9  |-  ( ( ( ( A  \  { X } )  ~~  x  /\  ( ( A 
\  { X }
)  i^i  { X } )  =  (/) )  /\  ( { X }  ~~  { x }  /\  ( x  i^i  {
x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2320, 22mpanl2 664 . . . . . . . 8  |-  ( ( ( A  \  { X } )  ~~  x  /\  ( { X }  ~~  { x }  /\  ( x  i^i  { x } )  =  (/) ) )  ->  (
( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) )
2423expcom 426 . . . . . . 7  |-  ( ( { X }  ~~  { x }  /\  (
x  i^i  { x } )  =  (/) )  ->  ( ( A 
\  { X }
)  ~~  x  ->  ( ( A  \  { X } )  u.  { X } )  ~~  (
x  u.  { x } ) ) )
2514, 17, 24syl2an 465 . . . . . 6  |-  ( ( X  e.  A  /\  x  e.  om )  ->  ( ( A  \  { X } )  ~~  x  ->  ( ( A 
\  { X }
)  u.  { X } )  ~~  (
x  u.  { x } ) ) )
26253ad2antl3 1122 . . . . 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 3946 . . . . . . . . 9  |-  ( X  e.  A  ->  (
( A  \  { X } )  u.  { X } )  =  A )
28 df-suc 4589 . . . . . . . . . . 11  |-  suc  x  =  ( x  u. 
{ x } )
2928eqcomi 2442 . . . . . . . . . 10  |-  ( x  u.  { x }
)  =  suc  x
3029a1i 11 . . . . . . . . 9  |-  ( X  e.  A  ->  (
x  u.  { x } )  =  suc  x )
3127, 30breq12d 4227 . . . . . . . 8  |-  ( X  e.  A  ->  (
( ( A  \  { X } )  u. 
{ X } ) 
~~  ( x  u. 
{ x } )  <-> 
A  ~~  suc  x ) )
32313ad2ant3 981 . . . . . . 7  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( ( ( A 
\  { X }
)  u.  { X } )  ~~  (
x  u.  { x } )  <->  A  ~~  suc  x ) )
3332adantr 453 . . . . . 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 7158 . . . . . . . . . . 11  |-  ( A 
~~  suc  M  ->  suc 
M  ~~  A )
35 entr 7161 . . . . . . . . . . . . 13  |-  ( ( suc  M  ~~  A  /\  A  ~~  suc  x
)  ->  suc  M  ~~  suc  x )
36 peano2 4867 . . . . . . . . . . . . . 14  |-  ( x  e.  om  ->  suc  x  e.  om )
37 nneneq 7292 . . . . . . . . . . . . . 14  |-  ( ( suc  M  e.  om  /\ 
suc  x  e.  om )  ->  ( suc  M  ~~  suc  x  <->  suc  M  =  suc  x ) )
3836, 37sylan2 462 . . . . . . . . . . . . 13  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( suc  M  ~~  suc  x  <->  suc  M  =  suc  x ) )
3935, 38syl5ib 212 . . . . . . . . . . . 12  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( ( suc  M  ~~  A  /\  A  ~~  suc  x )  ->  suc  M  =  suc  x ) )
4039exp3a 427 . . . . . . . . . . 11  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( suc  M  ~~  A  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) ) )
4134, 40syl5 31 . . . . . . . . . 10  |-  ( ( suc  M  e.  om  /\  x  e.  om )  ->  ( A  ~~  suc  M  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) ) )
421, 41sylan 459 . . . . . . . . 9  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( A  ~~  suc  M  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) ) )
4342imp 420 . . . . . . . 8  |-  ( ( ( M  e.  om  /\  x  e.  om )  /\  A  ~~  suc  M
)  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
4443an32s 781 . . . . . . 7  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M
)  /\  x  e.  om )  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
45443adantl3 1116 . . . . . 6  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( A  ~~  suc  x  ->  suc  M  =  suc  x ) )
4633, 45sylbid 208 . . . . 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 4869 . . . . . . 7  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( suc  M  =  suc  x  <->  M  =  x ) )
4847biimpd 200 . . . . . 6  |-  ( ( M  e.  om  /\  x  e.  om )  ->  ( suc  M  =  suc  x  ->  M  =  x ) )
49483ad2antl1 1120 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( suc  M  =  suc  x  ->  M  =  x )
)
5026, 46, 493syld 54 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( ( A  \  { X }
)  ~~  x  ->  M  =  x ) )
51 breq2 4218 . . . . 5  |-  ( M  =  x  ->  (
( A  \  { X } )  ~~  M  <->  ( A  \  { X } )  ~~  x
) )
5251biimprcd 218 . . . 4  |-  ( ( A  \  { X } )  ~~  x  ->  ( M  =  x  ->  ( A  \  { X } )  ~~  M ) )
5350, 52sylcom 28 . . 3  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  x  e.  om )  ->  ( ( A  \  { X }
)  ~~  x  ->  ( A  \  { X } )  ~~  M
) )
5453rexlimdva 2832 . 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708   _Vcvv 2958    \ cdif 3319    u. cun 3320    i^i cin 3321   (/)c0 3630   {csn 3816   class class class wbr 4214   Ord word 4582   suc csuc 4585   omcom 4847    ~~ cen 7108   Fincfn 7111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1o 6726  df-er 6907  df-en 7112  df-fin 7115
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