Theorem pwuninel2 6280

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 4158	. 2 |- ( U. A e. V -> -. ~P U. A C_ U. A )
2		elssuni 3837	. 2 |- ( ~P U. A e. A -> ~P U. A C_ U. A )
3	1, 2	nsyl 628	1 |- ( U. A e. V -> -. ~P U. A e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2148   C_ wss 3129   ~Pcpw 3575   U.cuni 3809

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4120

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-nel 2443  df-rab 2464  df-v 2739  df-in 3135  df-ss 3142  df-pw 3577  df-uni 3810

This theorem is referenced by:  pnfnre  7995