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Theorem rabsssn 4002
Description: Conditions for a restricted class abstraction to be a subset of a singleton, i.e. to be a singleton or the empty set. (Contributed by AV, 18-Apr-2019.)
Assertion
Ref Expression
rabsssn  |-  ( { x  e.  V  |  ph }  C_  { X } 
<-> 
A. x  e.  V  ( ph  ->  x  =  X ) )
Distinct variable group:    x, X
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem rabsssn
StepHypRef Expression
1 df-rab 2746 . . 3  |-  { x  e.  V  |  ph }  =  { x  |  ( x  e.  V  /\  ph ) }
2 df-sn 3969 . . 3  |-  { X }  =  { x  |  x  =  X }
31, 2sseq12i 3458 . 2  |-  ( { x  e.  V  |  ph }  C_  { X } 
<->  { x  |  ( x  e.  V  /\  ph ) }  C_  { x  |  x  =  X } )
4 ss2ab 3497 . 2  |-  ( { x  |  ( x  e.  V  /\  ph ) }  C_  { x  |  x  =  X } 
<-> 
A. x ( ( x  e.  V  /\  ph )  ->  x  =  X ) )
5 impexp 448 . . . 4  |-  ( ( ( x  e.  V  /\  ph )  ->  x  =  X )  <->  ( x  e.  V  ->  ( ph  ->  x  =  X ) ) )
65albii 1691 . . 3  |-  ( A. x ( ( x  e.  V  /\  ph )  ->  x  =  X )  <->  A. x ( x  e.  V  ->  ( ph  ->  x  =  X ) ) )
7 df-ral 2742 . . 3  |-  ( A. x  e.  V  ( ph  ->  x  =  X )  <->  A. x ( x  e.  V  ->  ( ph  ->  x  =  X ) ) )
86, 7bitr4i 256 . 2  |-  ( A. x ( ( x  e.  V  /\  ph )  ->  x  =  X )  <->  A. x  e.  V  ( ph  ->  x  =  X ) )
93, 4, 83bitri 275 1  |-  ( { x  e.  V  |  ph }  C_  { X } 
<-> 
A. x  e.  V  ( ph  ->  x  =  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1442    = wceq 1444    e. wcel 1887   {cab 2437   A.wral 2737   {crab 2741    C_ wss 3404   {csn 3968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rab 2746  df-in 3411  df-ss 3418  df-sn 3969
This theorem is referenced by:  suppmptcfin  40217  linc1  40271
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