Theorem pwuninel2 6340

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 4192	. 2 |- ( U. A e. V -> -. ~P U. A C_ U. A )
2		elssuni 3867	. 2 |- ( ~P U. A e. A -> ~P U. A C_ U. A )
3	1, 2	nsyl 629	1 |- ( U. A e. V -> -. ~P U. A e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2167   C_ wss 3157   ~Pcpw 3605   U.cuni 3839

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-sep 4151

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-nel 2463  df-rab 2484  df-v 2765  df-in 3163  df-ss 3170  df-pw 3607  df-uni 3840

This theorem is referenced by:  pnfnre  8068