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Theorem mreincl 13829
Description: Two closed sets have a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
mreincl  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  ( A  i^i  B )  e.  C )

Proof of Theorem mreincl
StepHypRef Expression
1 intprg 4086 . . 3  |-  ( ( A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
213adant1 976 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
3 simp1 958 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  C  e.  (Moore `  X )
)
4 prssi 3956 . . . 4  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
543adant1 976 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C
)
6 prnzg 3926 . . . 4  |-  ( A  e.  C  ->  { A ,  B }  =/=  (/) )
763ad2ant2 980 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  { A ,  B }  =/=  (/) )
8 mreintcl 13825 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  { A ,  B }  C_  C  /\  { A ,  B }  =/=  (/) )  ->  |^| { A ,  B }  e.  C )
93, 5, 7, 8syl3anc 1185 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  e.  C
)
102, 9eqeltrrd 2513 1  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  ( A  i^i  B )  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601    i^i cin 3321    C_ wss 3322   (/)c0 3630   {cpr 3817   |^|cint 4052   ` cfv 5457  Moorecmre 13812
This theorem is referenced by:  submacs  14770  subgacs  14980  nsgacs  14981  lsmmod  15312  lssacs  16048  mreclatdemo  17165  subrgacs  27499  sdrgacs  27500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-mre 13816
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