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Theorem shincli 26053
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 3123 . . 3  |-  A  e. 
_V
3 shincl.2 . . . 4  |-  B  e.  SH
43elexi 3123 . . 3  |-  B  e. 
_V
52, 4intpr 4315 . 2  |-  |^| { A ,  B }  =  ( A  i^i  B )
61, 3pm3.2i 455 . . . . 5  |-  ( A  e.  SH  /\  B  e.  SH )
72, 4prss 4181 . . . . 5  |-  ( ( A  e.  SH  /\  B  e.  SH )  <->  { A ,  B }  C_  SH )
86, 7mpbi 208 . . . 4  |-  { A ,  B }  C_  SH
92prnz 4146 . . . 4  |-  { A ,  B }  =/=  (/)
108, 9pm3.2i 455 . . 3  |-  ( { A ,  B }  C_  SH  /\  { A ,  B }  =/=  (/) )
1110shintcli 26020 . 2  |-  |^| { A ,  B }  e.  SH
125, 11eqeltrri 2552 1  |-  ( A  i^i  B )  e.  SH
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    e. wcel 1767    =/= wne 2662    i^i cin 3475    C_ wss 3476   (/)c0 3785   {cpr 4029   |^|cint 4282   SHcsh 25618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-hilex 25689  ax-hfvadd 25690  ax-hv0cl 25693  ax-hfvmul 25695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6288  df-sh 25897
This theorem is referenced by:  shincl  26072  shmodsi  26080  shmodi  26081  5oalem1  26345  5oalem3  26347  5oalem5  26349  5oalem6  26350  5oai  26352  3oalem2  26354  3oalem6  26358  cdj3lem1  27126
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