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Theorem shincli 22856
Description: Closure of intersection of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
shincl.1  |-  A  e.  SH
shincl.2  |-  B  e.  SH
Assertion
Ref Expression
shincli  |-  ( A  i^i  B )  e.  SH

Proof of Theorem shincli
StepHypRef Expression
1 shincl.1 . . . 4  |-  A  e.  SH
21elexi 2957 . . 3  |-  A  e. 
_V
3 shincl.2 . . . 4  |-  B  e.  SH
43elexi 2957 . . 3  |-  B  e. 
_V
52, 4intpr 4075 . 2  |-  |^| { A ,  B }  =  ( A  i^i  B )
61, 3pm3.2i 442 . . . . 5  |-  ( A  e.  SH  /\  B  e.  SH )
72, 4prss 3944 . . . . 5  |-  ( ( A  e.  SH  /\  B  e.  SH )  <->  { A ,  B }  C_  SH )
86, 7mpbi 200 . . . 4  |-  { A ,  B }  C_  SH
92prnz 3915 . . . 4  |-  { A ,  B }  =/=  (/)
108, 9pm3.2i 442 . . 3  |-  ( { A ,  B }  C_  SH  /\  { A ,  B }  =/=  (/) )
1110shintcli 22823 . 2  |-  |^| { A ,  B }  e.  SH
125, 11eqeltrri 2506 1  |-  ( A  i^i  B )  e.  SH
Colors of variables: wff set class
Syntax hints:    /\ wa 359    e. wcel 1725    =/= wne 2598    i^i cin 3311    C_ wss 3312   (/)c0 3620   {cpr 3807   |^|cint 4042   SHcsh 22423
This theorem is referenced by:  shincl  22875  shmodsi  22883  shmodi  22884  5oalem1  23148  5oalem3  23150  5oalem5  23152  5oalem6  23153  5oai  23155  3oalem2  23157  3oalem6  23161  cdj3lem1  23929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-hilex 22494  ax-hfvadd 22495  ax-hv0cl 22498  ax-hfvmul 22500
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-sh 22701
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