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Theorem mreincl 14850
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 4316 . . 3  |-  ( ( A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
213adant1 1014 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
3 simp1 996 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  C  e.  (Moore `  X )
)
4 prssi 4183 . . . 4  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
543adant1 1014 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C
)
6 prnzg 4147 . . . 4  |-  ( A  e.  C  ->  { A ,  B }  =/=  (/) )
763ad2ant2 1018 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  { A ,  B }  =/=  (/) )
8 mreintcl 14846 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  { A ,  B }  C_  C  /\  { A ,  B }  =/=  (/) )  ->  |^| { A ,  B }  e.  C )
93, 5, 7, 8syl3anc 1228 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  e.  C
)
102, 9eqeltrrd 2556 1  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  ( A  i^i  B )  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    i^i cin 3475    C_ wss 3476   (/)c0 3785   {cpr 4029   |^|cint 4282   ` cfv 5586  Moorecmre 14833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-mre 14837
This theorem is referenced by:  submacs  15806  subgacs  16031  nsgacs  16032  lsmmod  16489  lssacs  17396  mreclatdemoBAD  19363  subrgacs  30754  sdrgacs  30755
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