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Theorem mreincl 15215
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 4264 . . 3  |-  ( ( A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
213adant1 1017 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
3 simp1 999 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  C  e.  (Moore `  X )
)
4 prssi 4130 . . . 4  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
543adant1 1017 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C
)
6 prnzg 4094 . . . 4  |-  ( A  e.  C  ->  { A ,  B }  =/=  (/) )
763ad2ant2 1021 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  { A ,  B }  =/=  (/) )
8 mreintcl 15211 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  { A ,  B }  C_  C  /\  { A ,  B }  =/=  (/) )  ->  |^| { A ,  B }  e.  C )
93, 5, 7, 8syl3anc 1232 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  e.  C
)
102, 9eqeltrrd 2493 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 976    = wceq 1407    e. wcel 1844    =/= wne 2600    i^i cin 3415    C_ wss 3416   (/)c0 3740   {cpr 3976   |^|cint 4229   ` cfv 5571  Moorecmre 15198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-int 4230  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-iota 5535  df-fun 5573  df-fv 5579  df-mre 15202
This theorem is referenced by:  submacs  16322  subgacs  16562  nsgacs  16563  lsmmod  17019  lssacs  17935  mreclatdemoBAD  19892  subrgacs  35526  sdrgacs  35527
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