Theorem 2pwuninelg 6274

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 4547	. 2 |- -. ( A e. ~P ~P U. A /\ ~P ~P U. A e. A )
2		pwuni 4187	. . . 4 |- A C_ ~P U. A
3		elpwg 3580	. . . 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 664	1 |- ( A e. V -> -. ~P ~P U. A e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2146   C_ wss 3127   ~Pcpw 3572   U.cuni 3805

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-setind 4530

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-uni 3806

This theorem is referenced by:  mnfnre  7974