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Theorem nndifsnid 6403
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 3666 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 4519 . . . . . 6  |-  ( ( x  e.  A  /\  A  e.  om )  ->  x  e.  om )
21expcom 115 . . . . 5  |-  ( A  e.  om  ->  (
x  e.  A  ->  x  e.  om )
)
3 elnn 4519 . . . . . 6  |-  ( ( y  e.  A  /\  A  e.  om )  ->  y  e.  om )
43expcom 115 . . . . 5  |-  ( A  e.  om  ->  (
y  e.  A  -> 
y  e.  om )
)
52, 4anim12d 333 . . . 4  |-  ( A  e.  om  ->  (
( x  e.  A  /\  y  e.  A
)  ->  ( x  e.  om  /\  y  e. 
om ) ) )
6 nndceq 6395 . . . 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 2513 . 2  |-  ( A  e.  om  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y
)
9 dcdifsnid 6400 . 2  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  B  e.  A )  ->  (
( A  \  { B } )  u.  { B } )  =  A )
108, 9sylan 281 1  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( ( A  \  { B } )  u. 
{ B } )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103  DECID wdc 819    = wceq 1331    e. wcel 1480   A.wral 2416    \ cdif 3068    u. cun 3069   {csn 3527   omcom 4504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505
This theorem is referenced by:  phplem2  6747
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