Theorem 2pwuninelg 6429

Description: The power set of 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 Jim Kingdon, 14-Jan-2020.)

Assertion

Ref	Expression
2pwuninelg	|- ( A e. V -> -. ~P ~P U. A e. A )

Proof of Theorem 2pwuninelg

Step	Hyp	Ref	Expression
1		en2lp 4646	. 2 |- -. ( A e. ~P ~P U. A /\ ~P ~P U. A e. A )
2		pwuni 4276	. . . 4 |- A C_ ~P U. A
3		elpwg 3657	. . . 4 |- ( A e. V -> ( A e. ~P ~P U. A <-> A C_ ~P U. A ) )
4	2, 3	mpbiri 168	. . 3 |- ( A e. V -> A e. ~P ~P U. A )
5		ax-ia3 108	. . 3 |- ( A e. ~P ~P U. A -> ( ~P ~P U. A e. A -> ( A e. ~P ~P U. A /\ ~P ~P U. A e. A ) ) )
6	4, 5	syl 14	. 2 |- ( A e. V -> ( ~P ~P U. A e. A -> ( A e. ~P ~P U. A /\ ~P ~P U. A e. A ) ) )
7	1, 6	mtoi 668	1 |- ( A e. V -> -. ~P ~P U. A e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2200   C_ wss 3197   ~Pcpw 3649   U.cuni 3888

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889

This theorem is referenced by:  mnfnre  8189