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Theorem dfon2lem2 25413
Description: Lemma for dfon2 25421 (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 958 . . . 4  |-  ( ( x  C_  A  /\  ph 
/\  ps )  ->  x  C_  A )
21ss2abi 3417 . . 3  |-  { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  { x  |  x  C_  A }
3 df-pw 3803 . . 3  |-  ~P A  =  { x  |  x 
C_  A }
42, 3sseqtr4i 3383 . 2  |-  { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  ~P A
5 sspwuni 4178 . 2  |-  ( { x  |  ( x 
C_  A  /\  ph  /\ 
ps ) }  C_  ~P A  <->  U. { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  A )
64, 5mpbi 201 1  |-  U. {
x  |  ( x 
C_  A  /\  ph  /\ 
ps ) }  C_  A
Colors of variables: wff set class
Syntax hints:    /\ w3a 937   {cab 2424    C_ wss 3322   ~Pcpw 3801   U.cuni 4017
This theorem is referenced by:  dfon2lem3  25414  dfon2lem7  25418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-3an 939  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-pw 3803  df-uni 4018
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