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Theorem dfon2lem2 29433
Description: Lemma for dfon2 29441 (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 996 . . . 4  |-  ( ( x  C_  A  /\  ph 
/\  ps )  ->  x  C_  A )
21ss2abi 3568 . . 3  |-  { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  { x  |  x  C_  A }
3 df-pw 4017 . . 3  |-  ~P A  =  { x  |  x 
C_  A }
42, 3sseqtr4i 3532 . 2  |-  { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  ~P A
5 sspwuni 4421 . 2  |-  ( { x  |  ( x 
C_  A  /\  ph  /\ 
ps ) }  C_  ~P A  <->  U. { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  A )
64, 5mpbi 208 1  |-  U. {
x  |  ( x 
C_  A  /\  ph  /\ 
ps ) }  C_  A
Colors of variables: wff setvar class
Syntax hints:    /\ w3a 973   {cab 2442    C_ wss 3471   ~Pcpw 4015   U.cuni 4251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-in 3478  df-ss 3485  df-pw 4017  df-uni 4252
This theorem is referenced by:  dfon2lem3  29434  dfon2lem7  29438
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