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Theorem dif1en 6857
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 993 . . . 4  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  A  ~~  suc  M
)
21ensymd 6761 . . 3  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  suc  M  ~~  A
)
3 bren 6725 . . 3  |-  ( suc 
M  ~~  A  <->  E. f 
f : suc  M -1-1-onto-> A
)
42, 3sylib 121 . 2  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  E. f  f : suc  M -1-1-onto-> A )
5 peano2 4579 . . . . . . . 8  |-  ( M  e.  om  ->  suc  M  e.  om )
6 nnfi 6850 . . . . . . . 8  |-  ( suc 
M  e.  om  ->  suc 
M  e.  Fin )
75, 6syl 14 . . . . . . 7  |-  ( M  e.  om  ->  suc  M  e.  Fin )
873ad2ant1 1013 . . . . . 6  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  suc  M  e.  Fin )
9 enfii 6852 . . . . . 6  |-  ( ( suc  M  e.  Fin  /\  A  ~~  suc  M
)  ->  A  e.  Fin )
108, 1, 9syl2anc 409 . . . . 5  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  A  e.  Fin )
1110adantr 274 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  A  e.  Fin )
12 simpl3 997 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  X  e.  A )
13 f1of 5442 . . . . . 6  |-  ( f : suc  M -1-1-onto-> A  -> 
f : suc  M --> A )
1413adantl 275 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  f : suc  M --> A )
15 sucidg 4401 . . . . . . 7  |-  ( M  e.  om  ->  M  e.  suc  M )
16153ad2ant1 1013 . . . . . 6  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  M  e.  suc  M
)
1716adantr 274 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  M  e.  suc  M )
1814, 17ffvelrnd 5632 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  (
f `  M )  e.  A )
19 fidifsnen 6848 . . . 4  |-  ( ( A  e.  Fin  /\  X  e.  A  /\  ( f `  M
)  e.  A )  ->  ( A  \  { X } )  ~~  ( A  \  { ( f `  M ) } ) )
2011, 12, 18, 19syl3anc 1233 . . 3  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  ( A  \  { X }
)  ~~  ( A  \  { ( f `  M ) } ) )
21 nnord 4596 . . . . . . . 8  |-  ( M  e.  om  ->  Ord  M )
22 orddif 4531 . . . . . . . 8  |-  ( Ord 
M  ->  M  =  ( suc  M  \  { M } ) )
2321, 22syl 14 . . . . . . 7  |-  ( M  e.  om  ->  M  =  ( suc  M  \  { M } ) )
24233ad2ant1 1013 . . . . . 6  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  M  =  ( suc 
M  \  { M } ) )
2524adantr 274 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  M  =  ( suc  M  \  { M } ) )
2623eleq1d 2239 . . . . . . . . 9  |-  ( M  e.  om  ->  ( M  e.  om  <->  ( suc  M 
\  { M }
)  e.  om )
)
2726ibi 175 . . . . . . . 8  |-  ( M  e.  om  ->  ( suc  M  \  { M } )  e.  om )
28273ad2ant1 1013 . . . . . . 7  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( suc  M  \  { M } )  e. 
om )
2928adantr 274 . . . . . 6  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  ( suc  M  \  { M } )  e.  om )
30 dff1o2 5447 . . . . . . . . 9  |-  ( f : suc  M -1-1-onto-> A  <->  ( f  Fn  suc  M  /\  Fun  `' f  /\  ran  f  =  A ) )
3130simp2bi 1008 . . . . . . . 8  |-  ( f : suc  M -1-1-onto-> A  ->  Fun  `' f )
3231adantl 275 . . . . . . 7  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  Fun  `' f )
33 f1ofo 5449 . . . . . . . . 9  |-  ( f : suc  M -1-1-onto-> A  -> 
f : suc  M -onto-> A )
3433adantl 275 . . . . . . . 8  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  f : suc  M -onto-> A )
35 f1orel 5445 . . . . . . . . . . . 12  |-  ( f : suc  M -1-1-onto-> A  ->  Rel  f )
3635adantl 275 . . . . . . . . . . 11  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  Rel  f )
37 resdm 4930 . . . . . . . . . . 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 5446 . . . . . . . . . . . 12  |-  ( f : suc  M -1-1-onto-> A  ->  dom  f  =  suc  M )
4039reseq2d 4891 . . . . . . . . . . 11  |-  ( f : suc  M -1-1-onto-> A  -> 
( f  |`  dom  f
)  =  ( f  |`  suc  M ) )
4140adantl 275 . . . . . . . . . 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 2205 . . . . . . . . 9  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  f  =  ( f  |`  suc  M ) )
43 foeq1 5416 . . . . . . . . 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 146 . . . . . . 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 995 . . . . . . . . . 10  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  M  e.  om )
47 f1osng 5483 . . . . . . . . . 10  |-  ( ( M  e.  om  /\  ( f `  M
)  e.  A )  ->  { <. M , 
( f `  M
) >. } : { M } -1-1-onto-> { ( f `  M ) } )
4846, 18, 47syl2anc 409 . . . . . . . . 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 5449 . . . . . . . . 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 5443 . . . . . . . . . . 11  |-  ( f : suc  M -1-1-onto-> A  -> 
f  Fn  suc  M
)
5251adantl 275 . . . . . . . . . 10  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  f  Fn  suc  M )
53 fnressn 5682 . . . . . . . . . 10  |-  ( ( f  Fn  suc  M  /\  M  e.  suc  M )  ->  ( f  |` 
{ M } )  =  { <. M , 
( f `  M
) >. } )
5452, 17, 53syl2anc 409 . . . . . . . . 9  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  (
f  |`  { M }
)  =  { <. M ,  ( f `  M ) >. } )
55 foeq1 5416 . . . . . . . . 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 166 . . . . . . 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 5464 . . . . . . 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 1233 . . . . . 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 6735 . . . . . 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 409 . . . . 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 4011 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  M  ~~  ( A  \  {
( f `  M
) } ) )
6362ensymd 6761 . . 3  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  ( A  \  { ( f `
 M ) } )  ~~  M )
64 entr 6762 . . 3  |-  ( ( ( A  \  { X } )  ~~  ( A  \  { ( f `
 M ) } )  /\  ( A 
\  { ( f `
 M ) } )  ~~  M )  ->  ( A  \  { X } )  ~~  M )
6520, 63, 64syl2anc 409 . 2  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  ( A  \  { X }
)  ~~  M )
664, 65exlimddv 1891 1  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( A  \  { X } )  ~~  M
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 973    = wceq 1348   E.wex 1485    e. wcel 2141    \ cdif 3118   {csn 3583   <.cop 3586   class class class wbr 3989   Ord word 4347   suc csuc 4350   omcom 4574   `'ccnv 4610   dom cdm 4611   ran crn 4612    |` cres 4613   Rel wrel 4616   Fun wfun 5192    Fn wfn 5193   -->wf 5194   -onto->wfo 5196   -1-1-onto->wf1o 5197   ` cfv 5198    ~~ cen 6716   Fincfn 6718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-er 6513  df-en 6719  df-fin 6721
This theorem is referenced by:  dif1enen  6858  findcard  6866  findcard2  6867  findcard2s  6868  diffisn  6871  en2eleq  7172  en2other2  7173  zfz1isolem1  10775
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