Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfon2lem2 Unicode version

Theorem dfon2lem2 24140
Description: Lemma for dfon2 24148 (Contributed by Scott Fenton, 28-Feb-2011.)
Assertion
Ref Expression
dfon2lem2  |-  U. {
x  |  ( x 
C_  A  /\  ph  /\ 
ps ) }  C_  A
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem dfon2lem2
StepHypRef Expression
1 simp1 955 . . . 4  |-  ( ( x  C_  A  /\  ph 
/\  ps )  ->  x  C_  A )
21ss2abi 3245 . . 3  |-  { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  { x  |  x  C_  A }
3 df-pw 3627 . . 3  |-  ~P A  =  { x  |  x 
C_  A }
42, 3sseqtr4i 3211 . 2  |-  { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  ~P A
5 sspwuni 3987 . 2  |-  ( { x  |  ( x 
C_  A  /\  ph  /\ 
ps ) }  C_  ~P A  <->  U. { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  A )
64, 5mpbi 199 1  |-  U. {
x  |  ( x 
C_  A  /\  ph  /\ 
ps ) }  C_  A
Colors of variables: wff set class
Syntax hints:    /\ w3a 934   {cab 2269    C_ wss 3152   ~Pcpw 3625   U.cuni 3827
This theorem is referenced by:  dfon2lem3  24141  dfon2lem7  24145
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ral 2548  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-uni 3828
  Copyright terms: Public domain W3C validator