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Theorem rabrsn 4102
Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Proof shortened by AV, 21-Jul-2019.)
Assertion
Ref Expression
rabrsn  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( M  =  (/)  \/  M  =  { A } ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    M( x)

Proof of Theorem rabrsn
StepHypRef Expression
1 rabsnifsb 4100 . . 3  |-  { x  e.  { A }  |  ph }  =  if (
[. A  /  x ]. ph ,  { A } ,  (/) )
21eqeq2i 2475 . 2  |-  ( M  =  { x  e. 
{ A }  |  ph }  <->  M  =  if ( [. A  /  x ]. ph ,  { A } ,  (/) ) )
3 ifeqor 3988 . . . 4  |-  ( if ( [. A  /  x ]. ph ,  { A } ,  (/) )  =  { A }  \/  if ( [. A  /  x ]. ph ,  { A } ,  (/) )  =  (/) )
4 orcom 387 . . . 4  |-  ( ( if ( [. A  /  x ]. ph ,  { A } ,  (/) )  =  (/)  \/  if ( [. A  /  x ]. ph ,  { A } ,  (/) )  =  { A } )  <-> 
( if ( [. A  /  x ]. ph ,  { A } ,  (/) )  =  { A }  \/  if ( [. A  /  x ]. ph ,  { A } ,  (/) )  =  (/) ) )
53, 4mpbir 209 . . 3  |-  ( if ( [. A  /  x ]. ph ,  { A } ,  (/) )  =  (/)  \/  if ( [. A  /  x ]. ph ,  { A } ,  (/) )  =  { A } )
6 eqeq1 2461 . . . 4  |-  ( M  =  if ( [. A  /  x ]. ph ,  { A } ,  (/) )  ->  ( M  =  (/) 
<->  if ( [. A  /  x ]. ph ,  { A } ,  (/) )  =  (/) ) )
7 eqeq1 2461 . . . 4  |-  ( M  =  if ( [. A  /  x ]. ph ,  { A } ,  (/) )  ->  ( M  =  { A }  <->  if ( [. A  /  x ]. ph ,  { A } ,  (/) )  =  { A } ) )
86, 7orbi12d 709 . . 3  |-  ( M  =  if ( [. A  /  x ]. ph ,  { A } ,  (/) )  ->  ( ( M  =  (/)  \/  M  =  { A } )  <-> 
( if ( [. A  /  x ]. ph ,  { A } ,  (/) )  =  (/)  \/  if ( [. A  /  x ]. ph ,  { A } ,  (/) )  =  { A } ) ) )
95, 8mpbiri 233 . 2  |-  ( M  =  if ( [. A  /  x ]. ph ,  { A } ,  (/) )  ->  ( M  =  (/)  \/  M  =  { A } ) )
102, 9sylbi 195 1  |-  ( M  =  { x  e. 
{ A }  |  ph }  ->  ( M  =  (/)  \/  M  =  { A } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1395   {crab 2811   [.wsbc 3327   (/)c0 3793   ifcif 3944   {csn 4032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-nul 3794  df-if 3945  df-sn 4033
This theorem is referenced by:  hashrabrsn  12442  hashrabsn01  12443  hashrabsn1  12444  dvnprodlem3  31906
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