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Theorem shincli 24765
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 2982 . . 3  |-  A  e. 
_V
3 shincl.2 . . . 4  |-  B  e.  SH
43elexi 2982 . . 3  |-  B  e. 
_V
52, 4intpr 4161 . 2  |-  |^| { A ,  B }  =  ( A  i^i  B )
61, 3pm3.2i 455 . . . . 5  |-  ( A  e.  SH  /\  B  e.  SH )
72, 4prss 4027 . . . . 5  |-  ( ( A  e.  SH  /\  B  e.  SH )  <->  { A ,  B }  C_  SH )
86, 7mpbi 208 . . . 4  |-  { A ,  B }  C_  SH
92prnz 3994 . . . 4  |-  { A ,  B }  =/=  (/)
108, 9pm3.2i 455 . . 3  |-  ( { A ,  B }  C_  SH  /\  { A ,  B }  =/=  (/) )
1110shintcli 24732 . 2  |-  |^| { A ,  B }  e.  SH
125, 11eqeltrri 2514 1  |-  ( A  i^i  B )  e.  SH
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    e. wcel 1756    =/= wne 2606    i^i cin 3327    C_ wss 3328   (/)c0 3637   {cpr 3879   |^|cint 4128   SHcsh 24330
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-hilex 24401  ax-hfvadd 24402  ax-hv0cl 24405  ax-hfvmul 24407
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-csb 3289  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-iun 4173  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-ov 6094  df-sh 24609
This theorem is referenced by:  shincl  24784  shmodsi  24792  shmodi  24793  5oalem1  25057  5oalem3  25059  5oalem5  25061  5oalem6  25062  5oai  25064  3oalem2  25066  3oalem6  25070  cdj3lem1  25838
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