Theorem pwuninel2 6515

Description: The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (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 4274	. 2 |- ( U. A e. V -> -. ~P U. A C_ U. A )
2		elssuni 3944	. 2 |- ( ~P U. A e. A -> ~P U. A C_ U. A )
3	1, 2	nsyl 633	1 |- ( U. A e. V -> -. ~P U. A e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2205   C_ wss 3213   ~Pcpw 3671   U.cuni 3916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-sep 4230

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-nel 2510  df-rab 2531  df-v 2817  df-in 3219  df-ss 3226  df-pw 3673  df-uni 3917

This theorem is referenced by:  pnfnre  8317