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Theorem rabrsndc 3468
Description: A class abstraction over a decidable proposition restricted to a singleton is either the empty set or the singleton itself. (Contributed by Jim Kingdon, 8-Aug-2018.)
Hypotheses
Ref Expression
rabrsndc.1  |-  A  e. 
_V
rabrsndc.2  |- DECID  ph
Assertion
Ref Expression
rabrsndc  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( M  =  (/)  \/  M  =  { A } ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    M( x)

Proof of Theorem rabrsndc
StepHypRef Expression
1 rabrsndc.1 . . . . . 6  |-  A  e. 
_V
2 rabrsndc.2 . . . . . . . 8  |- DECID  ph
3 pm2.1dc 779 . . . . . . . 8  |-  (DECID  ph  ->  ( -.  ph  \/  ph )
)
42, 3ax-mp 7 . . . . . . 7  |-  ( -. 
ph  \/  ph )
54sbcth 2829 . . . . . 6  |-  ( A  e.  _V  ->  [. A  /  x ]. ( -. 
ph  \/  ph ) )
61, 5ax-mp 7 . . . . 5  |-  [. A  /  x ]. ( -. 
ph  \/  ph )
7 sbcor 2859 . . . . 5  |-  ( [. A  /  x ]. ( -.  ph  \/  ph )  <->  (
[. A  /  x ].  -.  ph  \/  [. A  /  x ]. ph )
)
86, 7mpbi 143 . . . 4  |-  ( [. A  /  x ].  -.  ph  \/  [. A  /  x ]. ph )
9 ralsns 3439 . . . . . 6  |-  ( A  e.  _V  ->  ( A. x  e.  { A }  -.  ph  <->  [. A  /  x ].  -.  ph ) )
101, 9ax-mp 7 . . . . 5  |-  ( A. x  e.  { A }  -.  ph  <->  [. A  /  x ].  -.  ph )
11 ralsns 3439 . . . . . 6  |-  ( A  e.  _V  ->  ( A. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
121, 11ax-mp 7 . . . . 5  |-  ( A. x  e.  { A } ph  <->  [. A  /  x ]. ph )
1310, 12orbi12i 714 . . . 4  |-  ( ( A. x  e.  { A }  -.  ph  \/  A. x  e.  { A } ph )  <->  ( [. A  /  x ].  -.  ph  \/  [. A  /  x ]. ph ) )
148, 13mpbir 144 . . 3  |-  ( A. x  e.  { A }  -.  ph  \/  A. x  e.  { A } ph )
15 rabeq0 3281 . . . 4  |-  ( { x  e.  { A }  |  ph }  =  (/)  <->  A. x  e.  { A }  -.  ph )
16 eqcom 2084 . . . . 5  |-  ( { x  e.  { A }  |  ph }  =  { A }  <->  { A }  =  { x  e.  { A }  |  ph } )
17 rabid2 2531 . . . . 5  |-  ( { A }  =  {
x  e.  { A }  |  ph }  <->  A. x  e.  { A } ph )
1816, 17bitri 182 . . . 4  |-  ( { x  e.  { A }  |  ph }  =  { A }  <->  A. x  e.  { A } ph )
1915, 18orbi12i 714 . . 3  |-  ( ( { x  e.  { A }  |  ph }  =  (/)  \/  { x  e.  { A }  |  ph }  =  { A } )  <->  ( A. x  e.  { A }  -.  ph  \/  A. x  e.  { A } ph ) )
2014, 19mpbir 144 . 2  |-  ( { x  e.  { A }  |  ph }  =  (/) 
\/  { x  e. 
{ A }  |  ph }  =  { A } )
21 eqeq1 2088 . . 3  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( M  =  (/)  <->  { x  e.  { A }  |  ph }  =  (/) ) )
22 eqeq1 2088 . . 3  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( M  =  { A }  <->  { x  e.  { A }  |  ph }  =  { A } ) )
2321, 22orbi12d 740 . 2  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( ( M  =  (/)  \/  M  =  { A } )  <-> 
( { x  e. 
{ A }  |  ph }  =  (/)  \/  {
x  e.  { A }  |  ph }  =  { A } ) ) )
2420, 23mpbiri 166 1  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( M  =  (/)  \/  M  =  { A } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    \/ wo 662  DECID wdc 776    = wceq 1285    e. wcel 1434   A.wral 2349   {crab 2353   _Vcvv 2602   [.wsbc 2816   (/)c0 3258   {csn 3406
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-nul 3259  df-sn 3412
This theorem is referenced by: (None)
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