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Theorem diffisn 6949
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 6815 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 120 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
32adantr 276 . 2  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  E. n  e.  om  A  ~~  n )
4 elex2 2776 . . . . . . . . 9  |-  ( B  e.  A  ->  E. x  x  e.  A )
54adantl 277 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  E. x  x  e.  A )
6 fin0 6941 . . . . . . . . 9  |-  ( A  e.  Fin  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
76adantr 276 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  ( A  =/=  (/)  <->  E. x  x  e.  A )
)
85, 7mpbird 167 . . . . . . 7  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  A  =/=  (/) )
98adantr 276 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  A  =/=  (/) )
109neneqd 2385 . . . . 5  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  -.  A  =  (/) )
11 simplrr 536 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  n
)
12 en0 6849 . . . . . . . . 9  |-  ( n 
~~  (/)  <->  n  =  (/) )
1312biimpri 133 . . . . . . . 8  |-  ( n  =  (/)  ->  n  ~~  (/) )
1413adantl 277 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  n  ~~  (/) )
15 entr 6838 . . . . . . 7  |-  ( ( A  ~~  n  /\  n  ~~  (/) )  ->  A  ~~  (/) )
1611, 14, 15syl2anc 411 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  (/) )
17 en0 6849 . . . . . 6  |-  ( A 
~~  (/)  <->  A  =  (/) )
1816, 17sylib 122 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  =  (/) )
1910, 18mtand 666 . . . 4  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  -.  n  =  (/) )
20 nn0suc 4636 . . . . . 6  |-  ( n  e.  om  ->  (
n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
2120orcomd 730 . . . . 5  |-  ( n  e.  om  ->  ( E. m  e.  om  n  =  suc  m  \/  n  =  (/) ) )
2221ad2antrl 490 . . . 4  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  -> 
( E. m  e. 
om  n  =  suc  m  \/  n  =  (/) ) )
2319, 22ecased 1360 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  E. m  e.  om  n  =  suc  m )
24 nnfi 6928 . . . . 5  |-  ( m  e.  om  ->  m  e.  Fin )
2524ad2antrl 490 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  m  e.  Fin )
26 simprl 529 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  m  e.  om )
27 simplrr 536 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  A  ~~  n )
28 breq2 4033 . . . . . . 7  |-  ( n  =  suc  m  -> 
( A  ~~  n  <->  A 
~~  suc  m )
)
2928ad2antll 491 . . . . . 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 147 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  A  ~~  suc  m )
31 simpllr 534 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  B  e.  A )
32 dif1en 6935 . . . . 5  |-  ( ( m  e.  om  /\  A  ~~  suc  m  /\  B  e.  A )  ->  ( A  \  { B } )  ~~  m
)
3326, 30, 31, 32syl3anc 1249 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  -> 
( A  \  { B } )  ~~  m
)
34 enfii 6930 . . . 4  |-  ( ( m  e.  Fin  /\  ( A  \  { B } )  ~~  m
)  ->  ( A  \  { B } )  e.  Fin )
3525, 33, 34syl2anc 411 . . 3  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  -> 
( A  \  { B } )  e.  Fin )
3623, 35rexlimddv 2616 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  -> 
( A  \  { B } )  e.  Fin )
373, 36rexlimddv 2616 1  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  ( A  \  { B } )  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1364   E.wex 1503    e. wcel 2164    =/= wne 2364   E.wrex 2473    \ cdif 3150   (/)c0 3446   {csn 3618   class class class wbr 4029   suc csuc 4396   omcom 4622    ~~ cen 6792   Fincfn 6794
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-er 6587  df-en 6795  df-fin 6797
This theorem is referenced by:  diffifi  6950  zfz1isolemsplit  10909  zfz1isolem1  10911  fsumdifsnconst  11598  fprodeq0g  11781
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