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Theorem dif1en 6881
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 998 . . . 4  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  A  ~~  suc  M
)
21ensymd 6785 . . 3  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  suc  M  ~~  A
)
3 bren 6749 . . 3  |-  ( suc 
M  ~~  A  <->  E. f 
f : suc  M -1-1-onto-> A
)
42, 3sylib 122 . 2  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  E. f  f : suc  M -1-1-onto-> A )
5 peano2 4596 . . . . . . . 8  |-  ( M  e.  om  ->  suc  M  e.  om )
6 nnfi 6874 . . . . . . . 8  |-  ( suc 
M  e.  om  ->  suc 
M  e.  Fin )
75, 6syl 14 . . . . . . 7  |-  ( M  e.  om  ->  suc  M  e.  Fin )
873ad2ant1 1018 . . . . . 6  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  suc  M  e.  Fin )
9 enfii 6876 . . . . . 6  |-  ( ( suc  M  e.  Fin  /\  A  ~~  suc  M
)  ->  A  e.  Fin )
108, 1, 9syl2anc 411 . . . . 5  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  A  e.  Fin )
1110adantr 276 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  A  e.  Fin )
12 simpl3 1002 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  X  e.  A )
13 f1of 5463 . . . . . 6  |-  ( f : suc  M -1-1-onto-> A  -> 
f : suc  M --> A )
1413adantl 277 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  f : suc  M --> A )
15 sucidg 4418 . . . . . . 7  |-  ( M  e.  om  ->  M  e.  suc  M )
16153ad2ant1 1018 . . . . . 6  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  M  e.  suc  M
)
1716adantr 276 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  M  e.  suc  M )
1814, 17ffvelcdmd 5654 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  (
f `  M )  e.  A )
19 fidifsnen 6872 . . . 4  |-  ( ( A  e.  Fin  /\  X  e.  A  /\  ( f `  M
)  e.  A )  ->  ( A  \  { X } )  ~~  ( A  \  { ( f `  M ) } ) )
2011, 12, 18, 19syl3anc 1238 . . 3  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  ( A  \  { X }
)  ~~  ( A  \  { ( f `  M ) } ) )
21 nnord 4613 . . . . . . . 8  |-  ( M  e.  om  ->  Ord  M )
22 orddif 4548 . . . . . . . 8  |-  ( Ord 
M  ->  M  =  ( suc  M  \  { M } ) )
2321, 22syl 14 . . . . . . 7  |-  ( M  e.  om  ->  M  =  ( suc  M  \  { M } ) )
24233ad2ant1 1018 . . . . . 6  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  M  =  ( suc 
M  \  { M } ) )
2524adantr 276 . . . . 5  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  M  =  ( suc  M  \  { M } ) )
2623eleq1d 2246 . . . . . . . . 9  |-  ( M  e.  om  ->  ( M  e.  om  <->  ( suc  M 
\  { M }
)  e.  om )
)
2726ibi 176 . . . . . . . 8  |-  ( M  e.  om  ->  ( suc  M  \  { M } )  e.  om )
28273ad2ant1 1018 . . . . . . 7  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( suc  M  \  { M } )  e. 
om )
2928adantr 276 . . . . . 6  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  ( suc  M  \  { M } )  e.  om )
30 dff1o2 5468 . . . . . . . . 9  |-  ( f : suc  M -1-1-onto-> A  <->  ( f  Fn  suc  M  /\  Fun  `' f  /\  ran  f  =  A ) )
3130simp2bi 1013 . . . . . . . 8  |-  ( f : suc  M -1-1-onto-> A  ->  Fun  `' f )
3231adantl 277 . . . . . . 7  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  Fun  `' f )
33 f1ofo 5470 . . . . . . . . 9  |-  ( f : suc  M -1-1-onto-> A  -> 
f : suc  M -onto-> A )
3433adantl 277 . . . . . . . 8  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  f : suc  M -onto-> A )
35 f1orel 5466 . . . . . . . . . . . 12  |-  ( f : suc  M -1-1-onto-> A  ->  Rel  f )
3635adantl 277 . . . . . . . . . . 11  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  Rel  f )
37 resdm 4948 . . . . . . . . . . 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 5467 . . . . . . . . . . . 12  |-  ( f : suc  M -1-1-onto-> A  ->  dom  f  =  suc  M )
4039reseq2d 4909 . . . . . . . . . . 11  |-  ( f : suc  M -1-1-onto-> A  -> 
( f  |`  dom  f
)  =  ( f  |`  suc  M ) )
4140adantl 277 . . . . . . . . . 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 2212 . . . . . . . . 9  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  f  =  ( f  |`  suc  M ) )
43 foeq1 5436 . . . . . . . . 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 147 . . . . . . 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 1000 . . . . . . . . . 10  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  M  e.  om )
47 f1osng 5504 . . . . . . . . . 10  |-  ( ( M  e.  om  /\  ( f `  M
)  e.  A )  ->  { <. M , 
( f `  M
) >. } : { M } -1-1-onto-> { ( f `  M ) } )
4846, 18, 47syl2anc 411 . . . . . . . . 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 5470 . . . . . . . . 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 5464 . . . . . . . . . . 11  |-  ( f : suc  M -1-1-onto-> A  -> 
f  Fn  suc  M
)
5251adantl 277 . . . . . . . . . 10  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  f  Fn  suc  M )
53 fnressn 5704 . . . . . . . . . 10  |-  ( ( f  Fn  suc  M  /\  M  e.  suc  M )  ->  ( f  |` 
{ M } )  =  { <. M , 
( f `  M
) >. } )
5452, 17, 53syl2anc 411 . . . . . . . . 9  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  (
f  |`  { M }
)  =  { <. M ,  ( f `  M ) >. } )
55 foeq1 5436 . . . . . . . . 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 167 . . . . . . 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 5485 . . . . . . 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 1238 . . . . . 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 6759 . . . . . 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 411 . . . . 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 4027 . . . 4  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  M  ~~  ( A  \  {
( f `  M
) } ) )
6362ensymd 6785 . . 3  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  ( A  \  { ( f `
 M ) } )  ~~  M )
64 entr 6786 . . 3  |-  ( ( ( A  \  { X } )  ~~  ( A  \  { ( f `
 M ) } )  /\  ( A 
\  { ( f `
 M ) } )  ~~  M )  ->  ( A  \  { X } )  ~~  M )
6520, 63, 64syl2anc 411 . 2  |-  ( ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A
)  /\  f : suc  M -1-1-onto-> A )  ->  ( A  \  { X }
)  ~~  M )
664, 65exlimddv 1898 1  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  X  e.  A )  ->  ( A  \  { X } )  ~~  M
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353   E.wex 1492    e. wcel 2148    \ cdif 3128   {csn 3594   <.cop 3597   class class class wbr 4005   Ord word 4364   suc csuc 4367   omcom 4591   `'ccnv 4627   dom cdm 4628   ran crn 4629    |` cres 4630   Rel wrel 4633   Fun wfun 5212    Fn wfn 5213   -->wf 5214   -onto->wfo 5216   -1-1-onto->wf1o 5217   ` cfv 5218    ~~ cen 6740   Fincfn 6742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-er 6537  df-en 6743  df-fin 6745
This theorem is referenced by:  dif1enen  6882  findcard  6890  findcard2  6891  findcard2s  6892  diffisn  6895  en2eleq  7196  en2other2  7197  zfz1isolem1  10822
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