Theorem bj-discrmoore 33066

Description: The discrete Moore collection on a set. (Contributed by BJ, 9-Dec-2021.)

Assertion

Ref	Expression
bj-discrmoore	|- ( A e. _V <-> ~P A e. Moore_ )

Proof of Theorem bj-discrmoore

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwexg 4850	. . 3 |- ( A e. _V -> ~P A e. _V )
2		unipw 4918	. . . . . 6 |- U. ~P A = A
3	2	ineq1i 3810	. . . . 5 |- ( U. ~P A i^i |^| x ) = ( A i^i |^| x )
4		inex1g 4801	. . . . . 6 |- ( A e. _V -> ( A i^i |^| x ) e. _V )
5		inss1 3833	. . . . . . 7 |- ( A i^i |^| x ) C_ A
6	5	a1i 11	. . . . . 6 |- ( A e. _V -> ( A i^i |^| x ) C_ A )
7	4, 6	elpwd 4167	. . . . 5 |- ( A e. _V -> ( A i^i |^| x ) e. ~P A )
8	3, 7	syl5eqel 2705	. . . 4 |- ( A e. _V -> ( U. ~P A i^i |^| x ) e. ~P A )
9	8	adantr 481	. . 3 |- ( ( A e. _V /\ x C_ ~P A ) -> ( U. ~P A i^i |^| x ) e. ~P A )
10	1, 9	bj-ismooredr 33064	. 2 |- ( A e. _V -> ~P A e. Moore_ )
11		pwexr 6974	. 2 |- ( ~P A e. Moore_ -> A e. _V )
12	10, 11	impbii 199	1 |- ( A e. _V <-> ~P A e. Moore_ )

Syntax hints:   <-> wb 196   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   |^|cint 4475  Moore_cmoore 33057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-bj-moore 33058

This theorem is referenced by: (None)