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Theorem dif1en 6337
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  f is distinct from all other variables.
StepHypRef Expression
1 simp2 905 . . . 4  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  A  ~~  suc  M
)
21ensymd 6263 . . 3  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  suc  M  ~~  A
)
3 bren 6228 . . 3  |-  ( suc 
M  ~~  A  <->  E. f 
f : suc  M -1-1-onto-> A
)
42, 3sylib 127 . 2  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  E. f  f : suc  M -1-1-onto-> A )
5 peano2 4318 . . . . . . . 8  |-  ( M  e.  om  ->  suc  M  e.  om )
6 nnfi 6333 . . . . . . . 8  |-  ( suc 
M  e.  om  ->  suc 
M  e.  Fin )
75, 6syl 14 . . . . . . 7  |-  ( M  e.  om  ->  suc  M  e.  Fin )
873ad2ant1 925 . . . . . 6  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  suc  M  e.  Fin )
9 enfii 6335 . . . . . 6  |-  ( ( suc  M  e.  Fin  /\  A  ~~  suc  M
)  ->  A  e.  Fin )
108, 1, 9syl2anc 391 . . . . 5  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  A  e.  Fin )
1110adantr 261 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  A  e.  Fin )
12 simpl3 909 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  X  e.  A )
13 f1of 5126 . . . . . 6  |-  ( f : suc  M -1-1-onto-> A  -> 
f : suc  M --> A )
1413adantl 262 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  f : suc  M --> A )
15 sucidg 4153 . . . . . . 7  |-  ( M  e.  om  ->  M  e.  suc  M )
16153ad2ant1 925 . . . . . 6  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  M  e.  suc  M
)
1716adantr 261 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  M  e.  suc  M )
1814, 17ffvelrnd 5303 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  (
f `  M )  e.  A )
19 fidifsnen 6331 . . . 4  |-  ( ( A  e.  Fin  /\  X  e.  A  /\  ( f `  M
)  e.  A )  ->  ( A  \  { X } )  ~~  ( A  \  { ( f `  M ) } ) )
2011, 12, 18, 19syl3anc 1135 . . 3  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  ( A  \  { X }
)  ~~  ( A  \  { ( f `  M ) } ) )
21 nnord 4334 . . . . . . . 8  |-  ( M  e.  om  ->  Ord  M )
22 orddif 4271 . . . . . . . 8  |-  ( Ord 
M  ->  M  =  ( suc  M  \  { M } ) )
2321, 22syl 14 . . . . . . 7  |-  ( M  e.  om  ->  M  =  ( suc  M  \  { M } ) )
24233ad2ant1 925 . . . . . 6  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  M  =  ( suc 
M  \  { M } ) )
2524adantr 261 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  M  =  ( suc  M  \  { M } ) )
2623eleq1d 2106 . . . . . . . . 9  |-  ( M  e.  om  ->  ( M  e.  om  <->  ( suc  M 
\  { M }
)  e.  om )
)
2726ibi 165 . . . . . . . 8  |-  ( M  e.  om  ->  ( suc  M  \  { M } )  e.  om )
28273ad2ant1 925 . . . . . . 7  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( suc  M  \  { M } )  e. 
om )
2928adantr 261 . . . . . 6  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  ( suc  M  \  { M } )  e.  om )
30 dff1o2 5131 . . . . . . . . 9  |-  ( f : suc  M -1-1-onto-> A  <->  ( f  Fn  suc  M  /\  Fun  `' f  /\  ran  f  =  A ) )
3130simp2bi 920 . . . . . . . 8  |-  ( f : suc  M -1-1-onto-> A  ->  Fun  `' f )
3231adantl 262 . . . . . . 7  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  Fun  `' f )
33 f1ofo 5133 . . . . . . . . 9  |-  ( f : suc  M -1-1-onto-> A  -> 
f : suc  M -onto-> A )
3433adantl 262 . . . . . . . 8  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  f : suc  M -onto-> A )
35 f1orel 5129 . . . . . . . . . . . 12  |-  ( f : suc  M -1-1-onto-> A  ->  Rel  f )
3635adantl 262 . . . . . . . . . . 11  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  Rel  f )
37 resdm 4649 . . . . . . . . . . 11  |-  ( Rel  f  ->  ( f  |` 
dom  f )  =  f )
3836, 37syl 14 . . . . . . . . . 10  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  (
f  |`  dom  f )  =  f )
39 f1odm 5130 . . . . . . . . . . . 12  |-  ( f : suc  M -1-1-onto-> A  ->  dom  f  =  suc  M )
4039reseq2d 4612 . . . . . . . . . . 11  |-  ( f : suc  M -1-1-onto-> A  -> 
( f  |`  dom  f
)  =  ( f  |`  suc  M ) )
4140adantl 262 . . . . . . . . . 10  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  (
f  |`  dom  f )  =  ( f  |`  suc  M ) )
4238, 41eqtr3d 2074 . . . . . . . . 9  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  f  =  ( f  |`  suc  M ) )
43 foeq1 5102 . . . . . . . . 9  |-  ( f  =  ( f  |`  suc  M )  ->  (
f : suc  M -onto-> A 
<->  ( f  |`  suc  M
) : suc  M -onto-> A ) )
4442, 43syl 14 . . . . . . . 8  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  (
f : suc  M -onto-> A 
<->  ( f  |`  suc  M
) : suc  M -onto-> A ) )
4534, 44mpbid 135 . . . . . . 7  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  (
f  |`  suc  M ) : suc  M -onto-> A
)
46 simpl1 907 . . . . . . . . . 10  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  M  e.  om )
47 f1osng 5167 . . . . . . . . . 10  |-  ( ( M  e.  om  /\  ( f `  M
)  e.  A )  ->  { <. M , 
( f `  M
) >. } : { M } -1-1-onto-> { ( f `  M ) } )
4846, 18, 47syl2anc 391 . . . . . . . . 9  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  { <. M ,  ( f `  M ) >. } : { M } -1-1-onto-> { ( f `  M ) } )
49 f1ofo 5133 . . . . . . . . 9  |-  ( {
<. M ,  ( f `
 M ) >. } : { M } -1-1-onto-> {
( f `  M
) }  ->  { <. M ,  ( f `  M ) >. } : { M } -onto-> { ( f `  M ) } )
5048, 49syl 14 . . . . . . . 8  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  { <. M ,  ( f `  M ) >. } : { M } -onto-> { ( f `  M ) } )
51 f1ofn 5127 . . . . . . . . . . 11  |-  ( f : suc  M -1-1-onto-> A  -> 
f  Fn  suc  M
)
5251adantl 262 . . . . . . . . . 10  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  f  Fn  suc  M )
53 fnressn 5349 . . . . . . . . . 10  |-  ( ( f  Fn  suc  M  /\  M  e.  suc  M )  ->  ( f  |` 
{ M } )  =  { <. M , 
( f `  M
) >. } )
5452, 17, 53syl2anc 391 . . . . . . . . 9  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  (
f  |`  { M }
)  =  { <. M ,  ( f `  M ) >. } )
55 foeq1 5102 . . . . . . . . 9  |-  ( ( f  |`  { M } )  =  { <. M ,  ( f `
 M ) >. }  ->  ( ( f  |`  { M } ) : { M } -onto-> { ( f `  M ) }  <->  { <. M , 
( f `  M
) >. } : { M } -onto-> { ( f `  M ) } ) )
5654, 55syl 14 . . . . . . . 8  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  (
( f  |`  { M } ) : { M } -onto-> { ( f `  M ) }  <->  { <. M , 
( f `  M
) >. } : { M } -onto-> { ( f `  M ) } ) )
5750, 56mpbird 156 . . . . . . 7  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  (
f  |`  { M }
) : { M } -onto-> { ( f `  M ) } )
58 resdif 5148 . . . . . . 7  |-  ( ( Fun  `' f  /\  ( f  |`  suc  M
) : suc  M -onto-> A  /\  ( f  |`  { M } ) : { M } -onto-> {
( f `  M
) } )  -> 
( f  |`  ( suc  M  \  { M } ) ) : ( suc  M  \  { M } ) -1-1-onto-> ( A 
\  { ( f `
 M ) } ) )
5932, 45, 57, 58syl3anc 1135 . . . . . 6  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  (
f  |`  ( suc  M  \  { M } ) ) : ( suc 
M  \  { M } ) -1-1-onto-> ( A  \  {
( f `  M
) } ) )
60 f1oeng 6237 . . . . . 6  |-  ( ( ( suc  M  \  { M } )  e. 
om  /\  ( f  |`  ( suc  M  \  { M } ) ) : ( suc  M  \  { M } ) -1-1-onto-> ( A  \  { ( f `  M ) } ) )  -> 
( suc  M  \  { M } )  ~~  ( A  \  { ( f `
 M ) } ) )
6129, 59, 60syl2anc 391 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  ( suc  M  \  { M } )  ~~  ( A  \  { ( f `
 M ) } ) )
6225, 61eqbrtrd 3784 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  M  ~~  ( A  \  {
( f `  M
) } ) )
6362ensymd 6263 . . 3  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  ( A  \  { ( f `
 M ) } )  ~~  M )
64 entr 6264 . . 3  |-  ( ( ( A  \  { X } )  ~~  ( A  \  { ( f `
 M ) } )  /\  ( A 
\  { ( f `
 M ) } )  ~~  M )  ->  ( A  \  { X } )  ~~  M )
6520, 63, 64syl2anc 391 . 2  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  ( A  \  { X }
)  ~~  M )
664, 65exlimddv 1778 1  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( A  \  { X } )  ~~  M
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    /\ w3a 885    = wceq 1243   E.wex 1381    e. wcel 1393    \ cdif 2914   {csn 3375   <.cop 3378   class class class wbr 3764   Ord word 4099   suc csuc 4102   omcom 4313   `'ccnv 4344   dom cdm 4345   ran crn 4346    |` cres 4347   Rel wrel 4350   Fun wfun 4896    Fn wfn 4897   -->wf 4898   -onto->wfo 4900   -1-1-onto->wf1o 4901   ` cfv 4902    ~~ cen 6219   Fincfn 6221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-er 6106  df-en 6222  df-fin 6224
This theorem is referenced by:  findcard  6345  findcard2  6346  findcard2s  6347  diffisn  6350
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