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Theorem diffisn 6791
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 6659 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 119 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
32adantr 274 . 2  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  E. n  e.  om  A  ~~  n )
4 elex2 2703 . . . . . . . . 9  |-  ( B  e.  A  ->  E. x  x  e.  A )
54adantl 275 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  E. x  x  e.  A )
6 fin0 6783 . . . . . . . . 9  |-  ( A  e.  Fin  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
76adantr 274 . . . . . . . 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 274 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  A  =/=  (/) )
109neneqd 2330 . . . . 5  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  -.  A  =  (/) )
11 simplrr 526 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  n
)
12 en0 6693 . . . . . . . . 9  |-  ( n 
~~  (/)  <->  n  =  (/) )
1312biimpri 132 . . . . . . . 8  |-  ( n  =  (/)  ->  n  ~~  (/) )
1413adantl 275 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  n  ~~  (/) )
15 entr 6682 . . . . . . 7  |-  ( ( A  ~~  n  /\  n  ~~  (/) )  ->  A  ~~  (/) )
1611, 14, 15syl2anc 409 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  (/) )
17 en0 6693 . . . . . 6  |-  ( A 
~~  (/)  <->  A  =  (/) )
1816, 17sylib 121 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  =  (/) )
1910, 18mtand 655 . . . 4  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  -.  n  =  (/) )
20 nn0suc 4522 . . . . . 6  |-  ( n  e.  om  ->  (
n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
2120orcomd 719 . . . . 5  |-  ( n  e.  om  ->  ( E. m  e.  om  n  =  suc  m  \/  n  =  (/) ) )
2221ad2antrl 482 . . . 4  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  -> 
( E. m  e. 
om  n  =  suc  m  \/  n  =  (/) ) )
2319, 22ecased 1328 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  E. m  e.  om  n  =  suc  m )
24 nnfi 6770 . . . . 5  |-  ( m  e.  om  ->  m  e.  Fin )
2524ad2antrl 482 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  m  e.  Fin )
26 simprl 521 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  m  e.  om )
27 simplrr 526 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  A  ~~  n )
28 breq2 3937 . . . . . . 7  |-  ( n  =  suc  m  -> 
( A  ~~  n  <->  A 
~~  suc  m )
)
2928ad2antll 483 . . . . . 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 524 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  B  e.  A )
32 dif1en 6777 . . . . 5  |-  ( ( m  e.  om  /\  A  ~~  suc  m  /\  B  e.  A )  ->  ( A  \  { B } )  ~~  m
)
3326, 30, 31, 32syl3anc 1217 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  -> 
( A  \  { B } )  ~~  m
)
34 enfii 6772 . . . 4  |-  ( ( m  e.  Fin  /\  ( A  \  { B } )  ~~  m
)  ->  ( A  \  { B } )  e.  Fin )
3525, 33, 34syl2anc 409 . . 3  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  -> 
( A  \  { B } )  e.  Fin )
3623, 35rexlimddv 2555 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  -> 
( A  \  { B } )  e.  Fin )
373, 36rexlimddv 2555 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 698    = wceq 1332   E.wex 1469    e. wcel 1481    =/= wne 2309   E.wrex 2418    \ cdif 3069   (/)c0 3364   {csn 3528   class class class wbr 3933   suc csuc 4291   omcom 4508    ~~ cen 6636   Fincfn 6638
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-if 3476  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-er 6433  df-en 6639  df-fin 6641
This theorem is referenced by:  diffifi  6792  zfz1isolemsplit  10609  zfz1isolem1  10611  fsumdifsnconst  11252
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