MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mressmrcd Structured version   Unicode version

Theorem mressmrcd 13852
Description: In a Moore system, if a set is between another set and its closure, the two sets have the same closure. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mressmrcd.1  |-  ( ph  ->  A  e.  (Moore `  X ) )
mressmrcd.2  |-  N  =  (mrCls `  A )
mressmrcd.3  |-  ( ph  ->  S  C_  ( N `  T ) )
mressmrcd.4  |-  ( ph  ->  T  C_  S )
Assertion
Ref Expression
mressmrcd  |-  ( ph  ->  ( N `  S
)  =  ( N `
 T ) )

Proof of Theorem mressmrcd
StepHypRef Expression
1 mressmrcd.1 . . . 4  |-  ( ph  ->  A  e.  (Moore `  X ) )
2 mressmrcd.2 . . . 4  |-  N  =  (mrCls `  A )
3 mressmrcd.3 . . . 4  |-  ( ph  ->  S  C_  ( N `  T ) )
41, 2mrcssvd 13848 . . . 4  |-  ( ph  ->  ( N `  T
)  C_  X )
51, 2, 3, 4mrcssd 13849 . . 3  |-  ( ph  ->  ( N `  S
)  C_  ( N `  ( N `  T
) ) )
6 mressmrcd.4 . . . . 5  |-  ( ph  ->  T  C_  S )
73, 4sstrd 3358 . . . . 5  |-  ( ph  ->  S  C_  X )
86, 7sstrd 3358 . . . 4  |-  ( ph  ->  T  C_  X )
91, 2, 8mrcidmd 13851 . . 3  |-  ( ph  ->  ( N `  ( N `  T )
)  =  ( N `
 T ) )
105, 9sseqtrd 3384 . 2  |-  ( ph  ->  ( N `  S
)  C_  ( N `  T ) )
111, 2, 6, 7mrcssd 13849 . 2  |-  ( ph  ->  ( N `  T
)  C_  ( N `  S ) )
1210, 11eqssd 3365 1  |-  ( ph  ->  ( N `  S
)  =  ( N `
 T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    C_ wss 3320   ` cfv 5454  Moorecmre 13807  mrClscmrc 13808
This theorem is referenced by:  mrieqvlemd  13854  mrissmrcd  13865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-mre 13811  df-mrc 13812
  Copyright terms: Public domain W3C validator