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Theorem diffisn 6555
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 6424 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 118 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
32adantr 270 . 2  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  E. n  e.  om  A  ~~  n )
4 elex2 2629 . . . . . . . . 9  |-  ( B  e.  A  ->  E. x  x  e.  A )
54adantl 271 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  E. x  x  e.  A )
6 fin0 6547 . . . . . . . . 9  |-  ( A  e.  Fin  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
76adantr 270 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  ( A  =/=  (/)  <->  E. x  x  e.  A )
)
85, 7mpbird 165 . . . . . . 7  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  A  =/=  (/) )
98adantr 270 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  A  =/=  (/) )
109neneqd 2272 . . . . 5  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  -.  A  =  (/) )
11 simplrr 503 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  n
)
12 en0 6458 . . . . . . . . 9  |-  ( n 
~~  (/)  <->  n  =  (/) )
1312biimpri 131 . . . . . . . 8  |-  ( n  =  (/)  ->  n  ~~  (/) )
1413adantl 271 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  n  ~~  (/) )
15 entr 6447 . . . . . . 7  |-  ( ( A  ~~  n  /\  n  ~~  (/) )  ->  A  ~~  (/) )
1611, 14, 15syl2anc 403 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  ~~  (/) )
17 en0 6458 . . . . . 6  |-  ( A 
~~  (/)  <->  A  =  (/) )
1816, 17sylib 120 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  n  =  (/) )  ->  A  =  (/) )
1910, 18mtand 624 . . . 4  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  -.  n  =  (/) )
20 nn0suc 4391 . . . . . 6  |-  ( n  e.  om  ->  (
n  =  (/)  \/  E. m  e.  om  n  =  suc  m ) )
2120orcomd 681 . . . . 5  |-  ( n  e.  om  ->  ( E. m  e.  om  n  =  suc  m  \/  n  =  (/) ) )
2221ad2antrl 474 . . . 4  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  -> 
( E. m  e. 
om  n  =  suc  m  \/  n  =  (/) ) )
2319, 22ecased 1283 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  ->  E. m  e.  om  n  =  suc  m )
24 nnfi 6534 . . . . 5  |-  ( m  e.  om  ->  m  e.  Fin )
2524ad2antrl 474 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  m  e.  Fin )
26 simprl 498 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  m  e.  om )
27 simplrr 503 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  A  ~~  n )
28 breq2 3824 . . . . . . 7  |-  ( n  =  suc  m  -> 
( A  ~~  n  <->  A 
~~  suc  m )
)
2928ad2antll 475 . . . . . 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 145 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  A  ~~  suc  m )
31 simpllr 501 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  ->  B  e.  A )
32 dif1en 6541 . . . . 5  |-  ( ( m  e.  om  /\  A  ~~  suc  m  /\  B  e.  A )  ->  ( A  \  { B } )  ~~  m
)
3326, 30, 31, 32syl3anc 1172 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  -> 
( A  \  { B } )  ~~  m
)
34 enfii 6536 . . . 4  |-  ( ( m  e.  Fin  /\  ( A  \  { B } )  ~~  m
)  ->  ( A  \  { B } )  e.  Fin )
3525, 33, 34syl2anc 403 . . 3  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  A )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  n  =  suc  m ) )  -> 
( A  \  { B } )  e.  Fin )
3623, 35rexlimddv 2489 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  A )  /\  ( n  e. 
om  /\  A  ~~  n ) )  -> 
( A  \  { B } )  e.  Fin )
373, 36rexlimddv 2489 1  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  ( A  \  { B } )  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1287   E.wex 1424    e. wcel 1436    =/= wne 2251   E.wrex 2356    \ cdif 2985   (/)c0 3275   {csn 3431   class class class wbr 3820   suc csuc 4165   omcom 4377    ~~ cen 6401   Fincfn 6403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3928  ax-sep 3931  ax-nul 3939  ax-pow 3983  ax-pr 4009  ax-un 4233  ax-setind 4325  ax-iinf 4375
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-if 3380  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3911  df-id 4093  df-iord 4166  df-on 4168  df-suc 4171  df-iom 4378  df-xp 4416  df-rel 4417  df-cnv 4418  df-co 4419  df-dm 4420  df-rn 4421  df-res 4422  df-ima 4423  df-iota 4943  df-fun 4980  df-fn 4981  df-f 4982  df-f1 4983  df-fo 4984  df-f1o 4985  df-fv 4986  df-er 6238  df-en 6404  df-fin 6406
This theorem is referenced by:  diffifi  6556  zfz1isolemsplit  10132  zfz1isolem1  10134
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