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Theorem shincli 22705
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 2901 . . 3  |-  A  e. 
_V
3 shincl.2 . . . 4  |-  B  e.  SH
43elexi 2901 . . 3  |-  B  e. 
_V
52, 4intpr 4018 . 2  |-  |^| { A ,  B }  =  ( A  i^i  B )
61, 3pm3.2i 442 . . . . 5  |-  ( A  e.  SH  /\  B  e.  SH )
72, 4prss 3888 . . . . 5  |-  ( ( A  e.  SH  /\  B  e.  SH )  <->  { A ,  B }  C_  SH )
86, 7mpbi 200 . . . 4  |-  { A ,  B }  C_  SH
92prnz 3859 . . . 4  |-  { A ,  B }  =/=  (/)
108, 9pm3.2i 442 . . 3  |-  ( { A ,  B }  C_  SH  /\  { A ,  B }  =/=  (/) )
1110shintcli 22672 . 2  |-  |^| { A ,  B }  e.  SH
125, 11eqeltrri 2451 1  |-  ( A  i^i  B )  e.  SH
Colors of variables: wff set class
Syntax hints:    /\ wa 359    e. wcel 1717    =/= wne 2543    i^i cin 3255    C_ wss 3256   (/)c0 3564   {cpr 3751   |^|cint 3985   SHcsh 22272
This theorem is referenced by:  shincl  22724  shmodsi  22732  shmodi  22733  5oalem1  22997  5oalem3  22999  5oalem5  23001  5oalem6  23002  5oai  23004  3oalem2  23006  3oalem6  23010  cdj3lem1  23778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-hilex 22343  ax-hfvadd 22344  ax-hv0cl 22347  ax-hfvmul 22349
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016  df-sh 22550
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