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Theorem nndifsnid 6371
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 3636 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 4489 . . . . . 6  |-  ( ( x  e.  A  /\  A  e.  om )  ->  x  e.  om )
21expcom 115 . . . . 5  |-  ( A  e.  om  ->  (
x  e.  A  ->  x  e.  om )
)
3 elnn 4489 . . . . . 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 6363 . . . 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 2490 . 2  |-  ( A  e.  om  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y
)
9 dcdifsnid 6368 . 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 804    = wceq 1316    e. wcel 1465   A.wral 2393    \ cdif 3038    u. cun 3039   {csn 3497   omcom 4474
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475
This theorem is referenced by:  phplem2  6715
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