Theorem ismred2 16263

Description: Properties that determine a Moore collection, using restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Hypotheses

Ref	Expression
ismred2.ss	|- ( ph -> C C_ ~P X )
ismred2.in	|- ( ( ph /\ s C_ C ) -> ( X i^i |^| s ) e. C )

Assertion

Ref	Expression
ismred2	|- ( ph -> C e. (Moore ` X ) )

Distinct variable groups:   ph, s   C, s   X, s

Proof of Theorem ismred2

Step	Hyp	Ref	Expression
1		ismred2.ss	. 2 |- ( ph -> C C_ ~P X )
2		eqid 2622	. . . 4 |- (/) = (/)
3		rint0 4517	. . . 4 |- ( (/) = (/) -> ( X i^i |^| (/) ) = X )
4	2, 3	ax-mp 5	. . 3 |- ( X i^i |^| (/) ) = X
5		0ss 3972	. . . 4 |- (/) C_ C
6		0ex 4790	. . . . 5 |- (/) e. _V
7		sseq1 3626	. . . . . . 7 |- ( s = (/) -> ( s C_ C <-> (/) C_ C ) )
8	7	anbi2d 740	. . . . . 6 |- ( s = (/) -> ( ( ph /\ s C_ C ) <-> ( ph /\ (/) C_ C ) ) )
9		inteq 4478	. . . . . . . 8 |- ( s = (/) -> |^| s = |^| (/) )
10	9	ineq2d 3814	. . . . . . 7 |- ( s = (/) -> ( X i^i |^| s ) = ( X i^i |^| (/) ) )
11	10	eleq1d 2686	. . . . . 6 |- ( s = (/) -> ( ( X i^i |^| s ) e. C <-> ( X i^i |^| (/) ) e. C ) )
12	8, 11	imbi12d 334	. . . . 5 |- ( s = (/) -> ( ( ( ph /\ s C_ C ) -> ( X i^i |^| s ) e. C ) <-> ( ( ph /\ (/) C_ C ) -> ( X i^i |^| (/) ) e. C ) ) )
13		ismred2.in	. . . . 5 |- ( ( ph /\ s C_ C ) -> ( X i^i |^| s ) e. C )
14	6, 12, 13	vtocl 3259	. . . 4 |- ( ( ph /\ (/) C_ C ) -> ( X i^i |^| (/) ) e. C )
15	5, 14	mpan2 707	. . 3 |- ( ph -> ( X i^i |^| (/) ) e. C )
16	4, 15	syl5eqelr 2706	. 2 |- ( ph -> X e. C )
17		simp2 1062	. . . . 5 |- ( ( ph /\ s C_ C /\ s =/= (/) ) -> s C_ C )
18	1	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ s C_ C /\ s =/= (/) ) -> C C_ ~P X )
19	17, 18	sstrd 3613	. . . 4 |- ( ( ph /\ s C_ C /\ s =/= (/) ) -> s C_ ~P X )
20		simp3 1063	. . . 4 |- ( ( ph /\ s C_ C /\ s =/= (/) ) -> s =/= (/) )
21		rintn0 4619	. . . 4 |- ( ( s C_ ~P X /\ s =/= (/) ) -> ( X i^i |^| s ) = |^| s )
22	19, 20, 21	syl2anc 693	. . 3 |- ( ( ph /\ s C_ C /\ s =/= (/) ) -> ( X i^i |^| s ) = |^| s )
23	13	3adant3 1081	. . 3 |- ( ( ph /\ s C_ C /\ s =/= (/) ) -> ( X i^i |^| s ) e. C )
24	22, 23	eqeltrrd 2702	. 2 |- ( ( ph /\ s C_ C /\ s =/= (/) ) -> |^| s e. C )
25	1, 16, 24	ismred 16262	1 |- ( ph -> C e. (Moore ` X ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   ` cfv 5888  Moorecmre 16242

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

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-mre 16246

This theorem is referenced by:  isacs1i  16318  mreacs  16319