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

Theorem ssintab 4069
Description: Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
ssintab  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  C_  x ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem ssintab
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssint 4068 . 2  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. y  e.  { x  |  ph } A  C_  y )
2 sseq2 3372 . . 3  |-  ( y  =  x  ->  ( A  C_  y  <->  A  C_  x
) )
32ralab2 3101 . 2  |-  ( A. y  e.  { x  |  ph } A  C_  y 
<-> 
A. x ( ph  ->  A  C_  x )
)
41, 3bitri 242 1  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  C_  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550   {cab 2424   A.wral 2707    C_ wss 3322   |^|cint 4052
This theorem is referenced by:  ssmin  4071  ssintrab  4075  intmin4  4081  dffi2  7431  rankval3b  7755  sstskm  8722  dfuzi  10365  cycsubg  14973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960  df-in 3329  df-ss 3336  df-int 4053
  Copyright terms: Public domain W3C validator