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Theorem pwuninel2 6299
Description: Direct proof of pwuninel 6300 avoiding functions and thus several ZF axioms. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
pwuninel2  |-  ( U. A  e.  V  ->  -. 
~P U. A  e.  A
)

Proof of Theorem pwuninel2
StepHypRef Expression
1 pwnss 4176 . 2  |-  ( U. A  e.  V  ->  -. 
~P U. A  C_  U. A
)
2 elssuni 3855 . 2  |-  ( ~P
U. A  e.  A  ->  ~P U. A  C_  U. A )
31, 2nsyl 113 1  |-  ( U. A  e.  V  ->  -. 
~P U. A  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   U.cuni 3827
This theorem is referenced by:  pwuninel  6300
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  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-nel 2449  df-rab 2552  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-uni 3828
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