Theorem 2pwuninelg 6514

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 4676	. 2 |- -. ( A e. ~P ~P U. A /\ ~P ~P U. A e. A )
2		pwuni 4305	. . . 4 |- A C_ ~P U. A
3		elpwg 3677	. . . 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 670	1 |- ( A e. V -> -. ~P ~P U. A e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2203   C_ wss 3211   ~Pcpw 3669   U.cuni 3914

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915

This theorem is referenced by:  mnfnre  8316