MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  intabs Unicode version

Theorem intabs 4172
Description: Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)
Hypotheses
Ref Expression
intabs.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
intabs.2  |-  ( x  =  |^| { y  |  ps }  ->  (
ph 
<->  ch ) )
intabs.3  |-  ( |^| { y  |  ps }  C_  A  /\  ch )
Assertion
Ref Expression
intabs  |-  |^| { x  |  ( x  C_  A  /\  ph ) }  =  |^| { x  |  ph }
Distinct variable groups:    x, y    x, A    ph, y    ps, x    ch, x
Allowed substitution hints:    ph( x)    ps( y)    ch( y)    A( y)

Proof of Theorem intabs
StepHypRef Expression
1 sseq1 3199 . . . . . 6  |-  ( x  =  |^| { y  |  ps }  ->  ( x  C_  A  <->  |^| { y  |  ps }  C_  A ) )
2 intabs.2 . . . . . 6  |-  ( x  =  |^| { y  |  ps }  ->  (
ph 
<->  ch ) )
31, 2anbi12d 691 . . . . 5  |-  ( x  =  |^| { y  |  ps }  ->  ( ( x  C_  A  /\  ph )  <->  ( |^| { y  |  ps }  C_  A  /\  ch )
) )
4 intabs.3 . . . . 5  |-  ( |^| { y  |  ps }  C_  A  /\  ch )
53, 4intmin3 3890 . . . 4  |-  ( |^| { y  |  ps }  e.  _V  ->  |^| { x  |  ( x  C_  A  /\  ph ) } 
C_  |^| { y  |  ps } )
6 intnex 4168 . . . . 5  |-  ( -. 
|^| { y  |  ps }  e.  _V  <->  |^| { y  |  ps }  =  _V )
7 ssv 3198 . . . . . 6  |-  |^| { x  |  ( x  C_  A  /\  ph ) } 
C_  _V
8 sseq2 3200 . . . . . 6  |-  ( |^| { y  |  ps }  =  _V  ->  ( |^| { x  |  ( x 
C_  A  /\  ph ) }  C_  |^| { y  |  ps }  <->  |^| { x  |  ( x  C_  A  /\  ph ) } 
C_  _V ) )
97, 8mpbiri 224 . . . . 5  |-  ( |^| { y  |  ps }  =  _V  ->  |^| { x  |  ( x  C_  A  /\  ph ) } 
C_  |^| { y  |  ps } )
106, 9sylbi 187 . . . 4  |-  ( -. 
|^| { y  |  ps }  e.  _V  ->  |^|
{ x  |  ( x  C_  A  /\  ph ) }  C_  |^| { y  |  ps } )
115, 10pm2.61i 156 . . 3  |-  |^| { x  |  ( x  C_  A  /\  ph ) } 
C_  |^| { y  |  ps }
12 intabs.1 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1312cbvabv 2402 . . . 4  |-  { x  |  ph }  =  {
y  |  ps }
1413inteqi 3866 . . 3  |-  |^| { x  |  ph }  =  |^| { y  |  ps }
1511, 14sseqtr4i 3211 . 2  |-  |^| { x  |  ( x  C_  A  /\  ph ) } 
C_  |^| { x  | 
ph }
16 simpr 447 . . . 4  |-  ( ( x  C_  A  /\  ph )  ->  ph )
1716ss2abi 3245 . . 3  |-  { x  |  ( x  C_  A  /\  ph ) } 
C_  { x  | 
ph }
18 intss 3883 . . 3  |-  ( { x  |  ( x 
C_  A  /\  ph ) }  C_  { x  |  ph }  ->  |^| { x  |  ph }  C_  |^| { x  |  ( x  C_  A  /\  ph ) } )
1917, 18ax-mp 8 . 2  |-  |^| { x  |  ph }  C_  |^| { x  |  ( x  C_  A  /\  ph ) }
2015, 19eqssi 3195 1  |-  |^| { x  |  ( x  C_  A  /\  ph ) }  =  |^| { x  |  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788    C_ wss 3152   |^|cint 3862
This theorem is referenced by:  dfnn3  9760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-int 3863
  Copyright terms: Public domain W3C validator