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Theorem pwuninel2 7006
Description: Direct proof of pwuninel 7007 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 4559 . 2  |-  ( U. A  e.  V  ->  -. 
~P U. A  C_  U. A
)
2 elssuni 4220 . 2  |-  ( ~P
U. A  e.  A  ->  ~P U. A  C_  U. A )
31, 2nsyl 121 1  |-  ( U. A  e.  V  ->  -. 
~P U. A  e.  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1842    C_ wss 3414   ~Pcpw 3955   U.cuni 4191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-nel 2601  df-rab 2763  df-v 3061  df-in 3421  df-ss 3428  df-pw 3957  df-uni 4192
This theorem is referenced by:  pwuninel  7007
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