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Theorem diffisn 6716
Description: Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.)
Assertion
Ref Expression
diffisn  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  ( A  \  { B } )  e.  Fin )

Proof of Theorem diffisn
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6585 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 119 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
32adantr 272 . 2  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  E. n  e.  om  A  ~~  n )
4 elex2 2657 . . . . . . . . 9  |-  ( B  e.  A  ->  E. x  x  e.  A )
54adantl 273 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  E. x  x  e.  A )
6 fin0 6708 . . . . . . . . 9  |-  ( A  e.  Fin  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
76adantr 272 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  ( A  =/=  (/)  <->  E. x  x  e.  A )
)
85, 7mpbird 166 . . . . . . 7  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  A  =/=  (/) )
98adantr 272 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  A  =/=  (/) )
109neneqd 2288 . . . . 5  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  -.  A  =  (/) )
11 simplrr 506 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  n
)
12 en0 6619 . . . . . . . . 9  |-  ( n 
~~  (/)  <->  n  =  (/) )
1312biimpri 132 . . . . . . . 8  |-  ( n  =  (/)  ->  n  ~~  (/) )
1413adantl 273 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  n  ~~  (/) )
15 entr 6608 . . . . . . 7  |-  ( ( A  ~~  n  /\  n  ~~  (/) )  ->  A  ~~  (/) )
1611, 14, 15syl2anc 406 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  (/) )
17 en0 6619 . . . . . 6  |-  ( A 
~~  (/)  <->  A  =  (/) )
1816, 17sylib 121 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  =  (/) )
1910, 18mtand 632 . . . 4  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  -.  n  =  (/) )
20 nn0suc 4456 . . . . . 6  |-  ( n  e.  om  ->  (
n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
2120orcomd 689 . . . . 5  |-  ( n  e.  om  ->  ( E. m  e.  om  n  =  suc  m  \/  n  =  (/) ) )
2221ad2antrl 477 . . . 4  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  -> 
( E. m  e. 
om  n  =  suc  m  \/  n  =  (/) ) )
2319, 22ecased 1295 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  E. m  e.  om  n  =  suc  m )
24 nnfi 6695 . . . . 5  |-  ( m  e.  om  ->  m  e.  Fin )
2524ad2antrl 477 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  m  e.  Fin )
26 simprl 501 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  m  e.  om )
27 simplrr 506 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  A  ~~  n )
28 breq2 3879 . . . . . . 7  |-  ( n  =  suc  m  -> 
( A  ~~  n  <->  A 
~~  suc  m )
)
2928ad2antll 478 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  -> 
( A  ~~  n  <->  A 
~~  suc  m )
)
3027, 29mpbid 146 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  A  ~~  suc  m )
31 simpllr 504 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  B  e.  A )
32 dif1en 6702 . . . . 5  |-  ( ( m  e.  om  /\  A  ~~  suc  m  /\  B  e.  A )  ->  ( A  \  { B } )  ~~  m
)
3326, 30, 31, 32syl3anc 1184 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  -> 
( A  \  { B } )  ~~  m
)
34 enfii 6697 . . . 4  |-  ( ( m  e.  Fin  /\  ( A  \  { B } )  ~~  m
)  ->  ( A  \  { B } )  e.  Fin )
3525, 33, 34syl2anc 406 . . 3  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  -> 
( A  \  { B } )  e.  Fin )
3623, 35rexlimddv 2513 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  -> 
( A  \  { B } )  e.  Fin )
373, 36rexlimddv 2513 1  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  ( A  \  { B } )  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 670    = wceq 1299   E.wex 1436    e. wcel 1448    =/= wne 2267   E.wrex 2376    \ cdif 3018   (/)c0 3310   {csn 3474   class class class wbr 3875   suc csuc 4225   omcom 4442    ~~ cen 6562   Fincfn 6564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-er 6359  df-en 6565  df-fin 6567
This theorem is referenced by:  diffifi  6717  zfz1isolemsplit  10422  zfz1isolem1  10424  fsumdifsnconst  11063
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