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Theorem dfon2lem2 24211
Description: Lemma for dfon2 24219 (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 3258 . . 3  |-  { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  { x  |  x  C_  A }
3 df-pw 3640 . . 3  |-  ~P A  =  { x  |  x 
C_  A }
42, 3sseqtr4i 3224 . 2  |-  { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  ~P A
5 sspwuni 4003 . 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 2282    C_ wss 3165   ~Pcpw 3638   U.cuni 3843
This theorem is referenced by:  dfon2lem3  24212  dfon2lem7  24216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844
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