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Theorem mreincl 14537
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 4162 . . 3  |-  ( ( A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
213adant1 1006 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
3 simp1 988 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  C  e.  (Moore `  X )
)
4 prssi 4029 . . . 4  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
543adant1 1006 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C
)
6 prnzg 3995 . . . 4  |-  ( A  e.  C  ->  { A ,  B }  =/=  (/) )
763ad2ant2 1010 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  { A ,  B }  =/=  (/) )
8 mreintcl 14533 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  { A ,  B }  C_  C  /\  { A ,  B }  =/=  (/) )  ->  |^| { A ,  B }  e.  C )
93, 5, 7, 8syl3anc 1218 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  |^| { A ,  B }  e.  C
)
102, 9eqeltrrd 2518 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606    i^i cin 3327    C_ wss 3328   (/)c0 3637   {cpr 3879   |^|cint 4128   ` cfv 5418  Moorecmre 14520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-mre 14524
This theorem is referenced by:  submacs  15493  subgacs  15716  nsgacs  15717  lsmmod  16172  lssacs  17048  mreclatdemoBAD  18700  subrgacs  29557  sdrgacs  29558
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