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Theorem mreincl 13711
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 3998 . . 3  |-  ( ( A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
213adant1 974 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
3 simp1 956 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  C  e.  (Moore `  X )
)
4 prssi 3869 . . . 4  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
543adant1 974 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C
)
6 prnzg 3839 . . . 4  |-  ( A  e.  C  ->  { A ,  B }  =/=  (/) )
763ad2ant2 978 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  { A ,  B }  =/=  (/) )
8 mreintcl 13707 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  { A ,  B }  C_  C  /\  { A ,  B }  =/=  (/) )  ->  |^| { A ,  B }  e.  C )
93, 5, 7, 8syl3anc 1183 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  e.  C
)
102, 9eqeltrrd 2441 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 935    = wceq 1647    e. wcel 1715    =/= wne 2529    i^i cin 3237    C_ wss 3238   (/)c0 3543   {cpr 3730   |^|cint 3964   ` cfv 5358  Moorecmre 13694
This theorem is referenced by:  submacs  14652  subgacs  14862  nsgacs  14863  lsmmod  15194  lssacs  15934  mreclatdemo  17050  subrgacs  27014  sdrgacs  27015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-int 3965  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-mre 13698
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