Theorem 2pwuninelg 6110

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 4407	. 2 |- -. ( A e. ~P ~P U. A /\ ~P ~P U. A e. A )
2		pwuni 4056	. . . 4 |- A C_ ~P U. A
3		elpwg 3465	. . . 4 |- ( A e. V -> ( A e. ~P ~P U. A <-> A C_ ~P U. A ) )
4	2, 3	mpbiri 167	. . 3 |- ( A e. V -> A e. ~P ~P U. A )
5		ax-ia3 107	. . 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 631	1 |- ( A e. V -> -. ~P ~P U. A e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 1448   C_ wss 3021   ~Pcpw 3457   U.cuni 3683

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-setind 4390

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684

This theorem is referenced by:  mnfnre  7680