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Theorem nndifsnid 6111
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 3538 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 variable  x is distinct from all other variables.
StepHypRef Expression
1 difsnss 3538 . . 3  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  C_  A
)
21adantl 266 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( ( A  \  { B } )  u. 
{ B } ) 
C_  A )
3 simpr 107 . . . . . . 7  |-  ( ( ( ( A  e. 
om  /\  B  e.  A )  /\  x  e.  A )  /\  x  =  B )  ->  x  =  B )
4 velsn 3420 . . . . . . 7  |-  ( x  e.  { B }  <->  x  =  B )
53, 4sylibr 141 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  A )  /\  x  e.  A )  /\  x  =  B )  ->  x  e.  { B } )
6 elun2 3139 . . . . . 6  |-  ( x  e.  { B }  ->  x  e.  ( ( A  \  { B } )  u.  { B } ) )
75, 6syl 14 . . . . 5  |-  ( ( ( ( A  e. 
om  /\  B  e.  A )  /\  x  e.  A )  /\  x  =  B )  ->  x  e.  ( ( A  \  { B } )  u. 
{ B } ) )
8 simplr 490 . . . . . . 7  |-  ( ( ( ( A  e. 
om  /\  B  e.  A )  /\  x  e.  A )  /\  -.  x  =  B )  ->  x  e.  A )
9 simpr 107 . . . . . . . 8  |-  ( ( ( ( A  e. 
om  /\  B  e.  A )  /\  x  e.  A )  /\  -.  x  =  B )  ->  -.  x  =  B )
109, 4sylnibr 612 . . . . . . 7  |-  ( ( ( ( A  e. 
om  /\  B  e.  A )  /\  x  e.  A )  /\  -.  x  =  B )  ->  -.  x  e.  { B } )
118, 10eldifd 2956 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  A )  /\  x  e.  A )  /\  -.  x  =  B )  ->  x  e.  ( A 
\  { B }
) )
12 elun1 3138 . . . . . 6  |-  ( x  e.  ( A  \  { B } )  ->  x  e.  ( ( A  \  { B }
)  u.  { B } ) )
1311, 12syl 14 . . . . 5  |-  ( ( ( ( A  e. 
om  /\  B  e.  A )  /\  x  e.  A )  /\  -.  x  =  B )  ->  x  e.  ( ( A  \  { B } )  u.  { B } ) )
14 simpr 107 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  ->  x  e.  A )
15 simpll 489 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  ->  A  e.  om )
16 elnn 4356 . . . . . . . 8  |-  ( ( x  e.  A  /\  A  e.  om )  ->  x  e.  om )
1714, 15, 16syl2anc 397 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  ->  x  e.  om )
18 simplr 490 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  ->  B  e.  A )
19 elnn 4356 . . . . . . . 8  |-  ( ( B  e.  A  /\  A  e.  om )  ->  B  e.  om )
2018, 15, 19syl2anc 397 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  ->  B  e.  om )
21 nndceq 6108 . . . . . . 7  |-  ( ( x  e.  om  /\  B  e.  om )  -> DECID  x  =  B )
2217, 20, 21syl2anc 397 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  -> DECID  x  =  B
)
23 df-dc 754 . . . . . 6  |-  (DECID  x  =  B  <->  ( x  =  B  \/  -.  x  =  B ) )
2422, 23sylib 131 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  ->  ( x  =  B  \/  -.  x  =  B )
)
257, 13, 24mpjaodan 722 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  ->  x  e.  ( ( A  \  { B } )  u. 
{ B } ) )
2625ex 112 . . 3  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( x  e.  A  ->  x  e.  ( ( A  \  { B } )  u.  { B } ) ) )
2726ssrdv 2979 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  A  C_  ( ( A  \  { B }
)  u.  { B } ) )
282, 27eqssd 2990 1  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( ( A  \  { B } )  u. 
{ B } )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    \/ wo 639  DECID wdc 753    = wceq 1259    e. wcel 1409    \ cdif 2942    u. cun 2943    C_ wss 2945   {csn 3403   omcom 4341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-int 3644  df-tr 3883  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342
This theorem is referenced by:  phplem2  6347
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