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Theorem mrcssvd 15115
Description: The Moore closure of a set is a subset of the base. Deduction form of mrcssv 15106. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mrcssd.1  |-  ( ph  ->  A  e.  (Moore `  X ) )
mrcssd.2  |-  N  =  (mrCls `  A )
Assertion
Ref Expression
mrcssvd  |-  ( ph  ->  ( N `  B
)  C_  X )

Proof of Theorem mrcssvd
StepHypRef Expression
1 mrcssd.1 . 2  |-  ( ph  ->  A  e.  (Moore `  X ) )
2 mrcssd.2 . . 3  |-  N  =  (mrCls `  A )
32mrcssv 15106 . 2  |-  ( A  e.  (Moore `  X
)  ->  ( N `  B )  C_  X
)
41, 3syl 16 1  |-  ( ph  ->  ( N `  B
)  C_  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823    C_ wss 3461   ` cfv 5570  Moorecmre 15074  mrClscmrc 15075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-mre 15078  df-mrc 15079
This theorem is referenced by:  mressmrcd  15119  mreexexlem2d  15137  mreacs  15150  acsmap2d  16011  gsumwspan  16216  cntzspan  17052  dprd2dlem1  17288  pgpfaclem2  17331  ismrcd2  30874
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