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Theorem diffisn 6892
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 6760 . . . 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 2753 . . . . . . . . 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 6884 . . . . . . . . 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 2368 . . . . 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 6794 . . . . . . . . 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 6783 . . . . . . 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 6794 . . . . . 6  |-  ( A 
~~  (/)  <->  A  =  (/) )
1816, 17sylib 122 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  =  (/) )
1910, 18mtand 665 . . . 4  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  -.  n  =  (/) )
20 nn0suc 4603 . . . . . 6  |-  ( n  e.  om  ->  (
n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
2120orcomd 729 . . . . 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 1349 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  E. m  e.  om  n  =  suc  m )
24 nnfi 6871 . . . . 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 4007 . . . . . . 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 6878 . . . . 5  |-  ( ( m  e.  om  /\  A  ~~  suc  m  /\  B  e.  A )  ->  ( A  \  { B } )  ~~  m
)
3326, 30, 31, 32syl3anc 1238 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  -> 
( A  \  { B } )  ~~  m
)
34 enfii 6873 . . . 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 2599 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  -> 
( A  \  { B } )  e.  Fin )
373, 36rexlimddv 2599 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 708    = wceq 1353   E.wex 1492    e. wcel 2148    =/= wne 2347   E.wrex 2456    \ cdif 3126   (/)c0 3422   {csn 3592   class class class wbr 4003   suc csuc 4365   omcom 4589    ~~ cen 6737   Fincfn 6739
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-er 6534  df-en 6740  df-fin 6742
This theorem is referenced by:  diffifi  6893  zfz1isolemsplit  10817  zfz1isolem1  10819  fsumdifsnconst  11462  fprodeq0g  11645
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