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Theorem fidifsnid 6870
Description: If we remove a single element from a finite set then put it back in, we end up with the original finite set. This strengthens difsnss 3738 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.)
Assertion
Ref Expression
fidifsnid  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  ( ( A  \  { B } )  u. 
{ B } )  =  A )

Proof of Theorem fidifsnid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fidceq 6868 . . . 4  |-  ( ( A  e.  Fin  /\  x  e.  A  /\  y  e.  A )  -> DECID  x  =  y )
213expb 1204 . . 3  |-  ( ( A  e.  Fin  /\  ( x  e.  A  /\  y  e.  A
) )  -> DECID  x  =  y
)
32ralrimivva 2559 . 2  |-  ( A  e.  Fin  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y
)
4 dcdifsnid 6504 . 2  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  B  e.  A )  ->  (
( A  \  { B } )  u.  { B } )  =  A )
53, 4sylan 283 1  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  ( ( A  \  { B } )  u. 
{ B } )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104  DECID wdc 834    = wceq 1353    e. wcel 2148   A.wral 2455    \ cdif 3126    u. cun 3127   {csn 3592   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-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-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-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  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-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-en 6740  df-fin 6742
This theorem is referenced by:  findcard2  6888  findcard2s  6889  xpfi  6928  fisseneq  6930  zfz1isolem1  10819
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