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Theorem pwuninel2 6511
Description: Direct proof of pwuninel 6512 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 4333 . 2  |-  ( U. A  e.  V  ->  -. 
~P U. A  C_  U. A
)
2 elssuni 4011 . 2  |-  ( ~P
U. A  e.  A  ->  ~P U. A  C_  U. A )
31, 2nsyl 115 1  |-  ( U. A  e.  V  ->  -. 
~P U. A  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1721    C_ wss 3288   ~Pcpw 3767   U.cuni 3983
This theorem is referenced by:  pwuninel  6512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-nel 2578  df-rab 2683  df-v 2926  df-in 3295  df-ss 3302  df-pw 3769  df-uni 3984
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