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Theorem nndifsnid 6080
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 3510 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 3510 . . 3  |-  ( B  e.  A  ->  (
( A  \  { B } )  u.  { B } )  C_  A
)
21adantl 262 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( ( A  \  { B } )  u. 
{ B } ) 
C_  A )
3 simpr 103 . . . . . . 7  |-  ( ( ( ( A  e. 
om  /\  B  e.  A )  /\  x  e.  A )  /\  x  =  B )  ->  x  =  B )
4 velsn 3392 . . . . . . 7  |-  ( x  e.  { B }  <->  x  =  B )
53, 4sylibr 137 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  A )  /\  x  e.  A )  /\  x  =  B )  ->  x  e.  { B } )
6 elun2 3111 . . . . . 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 482 . . . . . . 7  |-  ( ( ( ( A  e. 
om  /\  B  e.  A )  /\  x  e.  A )  /\  -.  x  =  B )  ->  x  e.  A )
9 simpr 103 . . . . . . . 8  |-  ( ( ( ( A  e. 
om  /\  B  e.  A )  /\  x  e.  A )  /\  -.  x  =  B )  ->  -.  x  =  B )
109, 4sylnibr 602 . . . . . . 7  |-  ( ( ( ( A  e. 
om  /\  B  e.  A )  /\  x  e.  A )  /\  -.  x  =  B )  ->  -.  x  e.  { B } )
118, 10eldifd 2928 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  A )  /\  x  e.  A )  /\  -.  x  =  B )  ->  x  e.  ( A 
\  { B }
) )
12 elun1 3110 . . . . . 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 103 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  ->  x  e.  A )
15 simpll 481 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  ->  A  e.  om )
16 elnn 4328 . . . . . . . 8  |-  ( ( x  e.  A  /\  A  e.  om )  ->  x  e.  om )
1714, 15, 16syl2anc 391 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  ->  x  e.  om )
18 simplr 482 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  ->  B  e.  A )
19 elnn 4328 . . . . . . . 8  |-  ( ( B  e.  A  /\  A  e.  om )  ->  B  e.  om )
2018, 15, 19syl2anc 391 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  ->  B  e.  om )
21 nndceq 6077 . . . . . . 7  |-  ( ( x  e.  om  /\  B  e.  om )  -> DECID  x  =  B )
2217, 20, 21syl2anc 391 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  -> DECID  x  =  B
)
23 df-dc 743 . . . . . 6  |-  (DECID  x  =  B  <->  ( x  =  B  \/  -.  x  =  B ) )
2422, 23sylib 127 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  ->  ( x  =  B  \/  -.  x  =  B )
)
257, 13, 24mpjaodan 711 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  A )  /\  x  e.  A
)  ->  x  e.  ( ( A  \  { B } )  u. 
{ B } ) )
2625ex 108 . . 3  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( x  e.  A  ->  x  e.  ( ( A  \  { B } )  u.  { B } ) ) )
2726ssrdv 2951 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  A  C_  ( ( A  \  { B }
)  u.  { B } ) )
282, 27eqssd 2962 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 97    \/ wo 629  DECID wdc 742    = wceq 1243    e. wcel 1393    \ cdif 2914    u. cun 2915    C_ wss 2917   {csn 3375   omcom 4313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314
This theorem is referenced by:  phplem2  6316
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