Theorem ismred 16262

Description: Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Hypotheses

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

Assertion

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

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

Proof of Theorem ismred

Step	Hyp	Ref	Expression
1		ismred.ss	. 2 |- ( ph -> C C_ ~P X )
2		ismred.ba	. 2 |- ( ph -> X e. C )
3		selpw 4165	. . . 4 |- ( s e. ~P C <-> s C_ C )
4		ismred.in	. . . . 5 |- ( ( ph /\ s C_ C /\ s =/= (/) ) -> |^| s e. C )
5	4	3expia 1267	. . . 4 |- ( ( ph /\ s C_ C ) -> ( s =/= (/) -> |^| s e. C ) )
6	3, 5	sylan2b 492	. . 3 |- ( ( ph /\ s e. ~P C ) -> ( s =/= (/) -> |^| s e. C ) )
7	6	ralrimiva 2966	. 2 |- ( ph -> A. s e. ~P C ( s =/= (/) -> |^| s e. C ) )
8		ismre 16250	. 2 |- ( C e. (Moore ` X ) <-> ( C C_ ~P X /\ X e. C /\ A. s e. ~P C ( s =/= (/) -> |^| s e. C ) ) )
9	1, 2, 7, 8	syl3anbrc 1246	1 |- ( ph -> C e. (Moore ` X ) )

Syntax hints:   -> wi 4   /\ w3a 1037   e. wcel 1990   =/= wne 2794   A.wral 2912   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-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:  ismred2  16263  mremre  16264  submre  16265  subrgmre  18804  lssmre  18966  cssmre  20037  cldmre  20882  toponmre  20897  ismrcd1  37261