ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rabrsndc Unicode version

Theorem rabrsndc 3660
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 837 . . . . . . . 8  |-  (DECID  ph  ->  ( -.  ph  \/  ph )
)
42, 3ax-mp 5 . . . . . . 7  |-  ( -. 
ph  \/  ph )
54sbcth 2976 . . . . . 6  |-  ( A  e.  _V  ->  [. A  /  x ]. ( -. 
ph  \/  ph ) )
61, 5ax-mp 5 . . . . 5  |-  [. A  /  x ]. ( -. 
ph  \/  ph )
7 sbcor 3007 . . . . 5  |-  ( [. A  /  x ]. ( -.  ph  \/  ph )  <->  (
[. A  /  x ].  -.  ph  \/  [. A  /  x ]. ph )
)
86, 7mpbi 145 . . . 4  |-  ( [. A  /  x ].  -.  ph  \/  [. A  /  x ]. ph )
9 ralsns 3630 . . . . . 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 3630 . . . . . 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 764 . . . 4  |-  ( ( A. x  e.  { A }  -.  ph  \/  A. x  e.  { A } ph )  <->  ( [. A  /  x ].  -.  ph  \/  [. A  /  x ]. ph ) )
148, 13mpbir 146 . . 3  |-  ( A. x  e.  { A }  -.  ph  \/  A. x  e.  { A } ph )
15 rabeq0 3452 . . . 4  |-  ( { x  e.  { A }  |  ph }  =  (/)  <->  A. x  e.  { A }  -.  ph )
16 eqcom 2179 . . . . 5  |-  ( { x  e.  { A }  |  ph }  =  { A }  <->  { A }  =  { x  e.  { A }  |  ph } )
17 rabid2 2653 . . . . 5  |-  ( { A }  =  {
x  e.  { A }  |  ph }  <->  A. x  e.  { A } ph )
1816, 17bitri 184 . . . 4  |-  ( { x  e.  { A }  |  ph }  =  { A }  <->  A. x  e.  { A } ph )
1915, 18orbi12i 764 . . 3  |-  ( ( { x  e.  { A }  |  ph }  =  (/)  \/  { x  e.  { A }  |  ph }  =  { A } )  <->  ( A. x  e.  { A }  -.  ph  \/  A. x  e.  { A } ph ) )
2014, 19mpbir 146 . 2  |-  ( { x  e.  { A }  |  ph }  =  (/) 
\/  { x  e. 
{ A }  |  ph }  =  { A } )
21 eqeq1 2184 . . 3  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( M  =  (/)  <->  { x  e.  { A }  |  ph }  =  (/) ) )
22 eqeq1 2184 . . 3  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( M  =  { A }  <->  { x  e.  { A }  |  ph }  =  { A } ) )
2321, 22orbi12d 793 . 2  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( ( M  =  (/)  \/  M  =  { A } )  <-> 
( { x  e. 
{ A }  |  ph }  =  (/)  \/  {
x  e.  { A }  |  ph }  =  { A } ) ) )
2420, 23mpbiri 168 1  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( M  =  (/)  \/  M  =  { A } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 105    \/ wo 708  DECID wdc 834    = wceq 1353    e. wcel 2148   A.wral 2455   {crab 2459   _Vcvv 2737   [.wsbc 2962   (/)c0 3422   {csn 3592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-nul 3423  df-sn 3598
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator