Theorem bj-ismooredr2 33065

Description: Sufficient condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)

Hypotheses

Ref	Expression
bj-ismooredr2.1	|- ( ph -> A e. V )
bj-ismooredr2.2	|- ( ph -> U. A e. A )
bj-ismooredr2.3	|- ( ( ( ph /\ x C_ A ) /\ x =/= (/) ) -> |^| x e. A )

Assertion

Ref	Expression
bj-ismooredr2	|- ( ph -> A e. Moore_ )

Distinct variable groups:   ph, x   x, A

Allowed substitution hint:   V( x)

Proof of Theorem bj-ismooredr2

Step	Hyp	Ref	Expression
1		selpw 4165	. . . . 5 |- ( x e. ~P A <-> x C_ A )
2		pm2.1 433	. . . . . . . 8 |- ( -. x = (/) \/ x = (/) )
3	2	biantru 526	. . . . . . 7 |- ( x C_ A <-> ( x C_ A /\ ( -. x = (/) \/ x = (/) ) ) )
4		andi 911	. . . . . . 7 |- ( ( x C_ A /\ ( -. x = (/) \/ x = (/) ) ) <-> ( ( x C_ A /\ -. x = (/) ) \/ ( x C_ A /\ x = (/) ) ) )
5	3, 4	bitri 264	. . . . . 6 |- ( x C_ A <-> ( ( x C_ A /\ -. x = (/) ) \/ ( x C_ A /\ x = (/) ) ) )
6		df-ne 2795	. . . . . . . . 9 |- ( x =/= (/) <-> -. x = (/) )
7	6	bicomi 214	. . . . . . . 8 |- ( -. x = (/) <-> x =/= (/) )
8	7	anbi2i 730	. . . . . . 7 |- ( ( x C_ A /\ -. x = (/) ) <-> ( x C_ A /\ x =/= (/) ) )
9		simpr 477	. . . . . . . 8 |- ( ( x C_ A /\ x = (/) ) -> x = (/) )
10		id 22	. . . . . . . . . 10 |- ( x = (/) -> x = (/) )
11		0ss 3972	. . . . . . . . . 10 |- (/) C_ A
12	10, 11	syl6eqss 3655	. . . . . . . . 9 |- ( x = (/) -> x C_ A )
13	12	ancri 575	. . . . . . . 8 |- ( x = (/) -> ( x C_ A /\ x = (/) ) )
14	9, 13	impbii 199	. . . . . . 7 |- ( ( x C_ A /\ x = (/) ) <-> x = (/) )
15	8, 14	orbi12i 543	. . . . . 6 |- ( ( ( x C_ A /\ -. x = (/) ) \/ ( x C_ A /\ x = (/) ) ) <-> ( ( x C_ A /\ x =/= (/) ) \/ x = (/) ) )
16	5, 15	bitri 264	. . . . 5 |- ( x C_ A <-> ( ( x C_ A /\ x =/= (/) ) \/ x = (/) ) )
17	1, 16	bitri 264	. . . 4 |- ( x e. ~P A <-> ( ( x C_ A /\ x =/= (/) ) \/ x = (/) ) )
18		bj-ismooredr2.3	. . . . . . 7 |- ( ( ( ph /\ x C_ A ) /\ x =/= (/) ) -> |^| x e. A )
19	18	expl 648	. . . . . 6 |- ( ph -> ( ( x C_ A /\ x =/= (/) ) -> |^| x e. A ) )
20		intssuni2 4502	. . . . . . 7 |- ( ( x C_ A /\ x =/= (/) ) -> |^| x C_ U. A )
21		sseqin2 3817	. . . . . . . . . . 11 |- ( |^| x C_ U. A <-> ( U. A i^i |^| x ) = |^| x )
22	21	biimpi 206	. . . . . . . . . 10 |- ( |^| x C_ U. A -> ( U. A i^i |^| x ) = |^| x )
23	22	eqcomd 2628	. . . . . . . . 9 |- ( |^| x C_ U. A -> |^| x = ( U. A i^i |^| x ) )
24	23	eleq1d 2686	. . . . . . . 8 |- ( |^| x C_ U. A -> ( |^| x e. A <-> ( U. A i^i |^| x ) e. A ) )
25	24	biimpd 219	. . . . . . 7 |- ( |^| x C_ U. A -> ( |^| x e. A -> ( U. A i^i |^| x ) e. A ) )
26	20, 25	syl 17	. . . . . 6 |- ( ( x C_ A /\ x =/= (/) ) -> ( |^| x e. A -> ( U. A i^i |^| x ) e. A ) )
27	19, 26	sylcom 30	. . . . 5 |- ( ph -> ( ( x C_ A /\ x =/= (/) ) -> ( U. A i^i |^| x ) e. A ) )
28		bj-ismooredr2.2	. . . . . 6 |- ( ph -> U. A e. A )
29		rint0 4517	. . . . . . . 8 |- ( x = (/) -> ( U. A i^i |^| x ) = U. A )
30	29	eqcomd 2628	. . . . . . 7 |- ( x = (/) -> U. A = ( U. A i^i |^| x ) )
31	30	eleq1d 2686	. . . . . 6 |- ( x = (/) -> ( U. A e. A <-> ( U. A i^i |^| x ) e. A ) )
32	28, 31	syl5ibcom 235	. . . . 5 |- ( ph -> ( x = (/) -> ( U. A i^i |^| x ) e. A ) )
33	27, 32	jaod 395	. . . 4 |- ( ph -> ( ( ( x C_ A /\ x =/= (/) ) \/ x = (/) ) -> ( U. A i^i |^| x ) e. A ) )
34	17, 33	syl5bi 232	. . 3 |- ( ph -> ( x e. ~P A -> ( U. A i^i |^| x ) e. A ) )
35	34	ralrimiv 2965	. 2 |- ( ph -> A. x e. ~P A ( U. A i^i |^| x ) e. A )
36		bj-ismooredr2.1	. . 3 |- ( ph -> A e. V )
37		bj-ismoore 33059	. . 3 |- ( A e. V -> ( A e. Moore_ <-> A. x e. ~P A ( U. A i^i |^| x ) e. A ) )
38	36, 37	syl 17	. 2 |- ( ph -> ( A e. Moore_ <-> A. x e. ~P A ( U. A i^i |^| x ) e. A ) )
39	35, 38	mpbird 247	1 |- ( ph -> A e. Moore_ )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-uni 4437  df-int 4476  df-bj-moore 33058

This theorem is referenced by:  bj-snmoore  33068