Theorem pwuninel2 7400

Description: Direct proof of pwuninel 7401 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

Step	Hyp	Ref	Expression
1		pwnss 4830	. 2 |- ( U. A e. V -> -. ~P U. A C_ U. A )
2		elssuni 4467	. 2 |- ( ~P U. A e. A -> ~P U. A C_ U. A )
3	1, 2	nsyl 135	1 |- ( U. A e. V -> -. ~P U. A e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437

This theorem is referenced by:  pwuninel  7401