Theorem pwuninel2 6179

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 4083	. 2 |- ( U. A e. V -> -. ~P U. A C_ U. A )
2		elssuni 3764	. 2 |- ( ~P U. A e. A -> ~P U. A C_ U. A )
3	1, 2	nsyl 617	1 |- ( U. A e. V -> -. ~P U. A e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1480   C_ wss 3071   ~Pcpw 3510   U.cuni 3736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-nel 2404  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737

This theorem is referenced by:  pnfnre  7807