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Theorem nndifsnid 6218
Description: If we remove a single element from a natural number then put it back in, we end up with the original natural number. This strengthens difsnss 3566 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.)
Assertion
Ref Expression
nndifsnid  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( ( A  \  { B } )  u. 
{ B } )  =  A )

Proof of Theorem nndifsnid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn 4393 . . . . . 6  |-  ( ( x  e.  A  /\  A  e.  om )  ->  x  e.  om )
21expcom 114 . . . . 5  |-  ( A  e.  om  ->  (
x  e.  A  ->  x  e.  om )
)
3 elnn 4393 . . . . . 6  |-  ( ( y  e.  A  /\  A  e.  om )  ->  y  e.  om )
43expcom 114 . . . . 5  |-  ( A  e.  om  ->  (
y  e.  A  -> 
y  e.  om )
)
52, 4anim12d 328 . . . 4  |-  ( A  e.  om  ->  (
( x  e.  A  /\  y  e.  A
)  ->  ( x  e.  om  /\  y  e. 
om ) ) )
6 nndceq 6214 . . . 4  |-  ( ( x  e.  om  /\  y  e.  om )  -> DECID  x  =  y )
75, 6syl6 33 . . 3  |-  ( A  e.  om  ->  (
( x  e.  A  /\  y  e.  A
)  -> DECID  x  =  y
) )
87ralrimivv 2450 . 2  |-  ( A  e.  om  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y
)
9 dcdifsnid 6217 . 2  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  B  e.  A )  ->  (
( A  \  { B } )  u.  { B } )  =  A )
108, 9sylan 277 1  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( ( A  \  { B } )  u. 
{ B } )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102  DECID wdc 778    = wceq 1287    e. wcel 1436   A.wral 2355    \ cdif 2985    u. cun 2986   {csn 3431   omcom 4378
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-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-uni 3637  df-int 3672  df-tr 3912  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379
This theorem is referenced by:  phplem2  6521
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