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Theorem rabrsndc 3561
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 807 . . . . . . . 8  |-  (DECID  ph  ->  ( -.  ph  \/  ph )
)
42, 3ax-mp 5 . . . . . . 7  |-  ( -. 
ph  \/  ph )
54sbcth 2895 . . . . . 6  |-  ( A  e.  _V  ->  [. A  /  x ]. ( -. 
ph  \/  ph ) )
61, 5ax-mp 5 . . . . 5  |-  [. A  /  x ]. ( -. 
ph  \/  ph )
7 sbcor 2925 . . . . 5  |-  ( [. A  /  x ]. ( -.  ph  \/  ph )  <->  (
[. A  /  x ].  -.  ph  \/  [. A  /  x ]. ph )
)
86, 7mpbi 144 . . . 4  |-  ( [. A  /  x ].  -.  ph  \/  [. A  /  x ]. ph )
9 ralsns 3532 . . . . . 6  |-  ( A  e.  _V  ->  ( A. x  e.  { A }  -.  ph  <->  [. A  /  x ].  -.  ph ) )
101, 9ax-mp 5 . . . . 5  |-  ( A. x  e.  { A }  -.  ph  <->  [. A  /  x ].  -.  ph )
11 ralsns 3532 . . . . . 6  |-  ( A  e.  _V  ->  ( A. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
121, 11ax-mp 5 . . . . 5  |-  ( A. x  e.  { A } ph  <->  [. A  /  x ]. ph )
1310, 12orbi12i 738 . . . 4  |-  ( ( A. x  e.  { A }  -.  ph  \/  A. x  e.  { A } ph )  <->  ( [. A  /  x ].  -.  ph  \/  [. A  /  x ]. ph ) )
148, 13mpbir 145 . . 3  |-  ( A. x  e.  { A }  -.  ph  \/  A. x  e.  { A } ph )
15 rabeq0 3362 . . . 4  |-  ( { x  e.  { A }  |  ph }  =  (/)  <->  A. x  e.  { A }  -.  ph )
16 eqcom 2119 . . . . 5  |-  ( { x  e.  { A }  |  ph }  =  { A }  <->  { A }  =  { x  e.  { A }  |  ph } )
17 rabid2 2584 . . . . 5  |-  ( { A }  =  {
x  e.  { A }  |  ph }  <->  A. x  e.  { A } ph )
1816, 17bitri 183 . . . 4  |-  ( { x  e.  { A }  |  ph }  =  { A }  <->  A. x  e.  { A } ph )
1915, 18orbi12i 738 . . 3  |-  ( ( { x  e.  { A }  |  ph }  =  (/)  \/  { x  e.  { A }  |  ph }  =  { A } )  <->  ( A. x  e.  { A }  -.  ph  \/  A. x  e.  { A } ph ) )
2014, 19mpbir 145 . 2  |-  ( { x  e.  { A }  |  ph }  =  (/) 
\/  { x  e. 
{ A }  |  ph }  =  { A } )
21 eqeq1 2124 . . 3  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( M  =  (/)  <->  { x  e.  { A }  |  ph }  =  (/) ) )
22 eqeq1 2124 . . 3  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( M  =  { A }  <->  { x  e.  { A }  |  ph }  =  { A } ) )
2321, 22orbi12d 767 . 2  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( ( M  =  (/)  \/  M  =  { A } )  <-> 
( { x  e. 
{ A }  |  ph }  =  (/)  \/  {
x  e.  { A }  |  ph }  =  { A } ) ) )
2420, 23mpbiri 167 1  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( M  =  (/)  \/  M  =  { A } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    \/ wo 682  DECID wdc 804    = wceq 1316    e. wcel 1465   A.wral 2393   {crab 2397   _Vcvv 2660   [.wsbc 2882   (/)c0 3333   {csn 3497
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-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-dc 805  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-nul 3334  df-sn 3503
This theorem is referenced by: (None)
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