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Theorem shincli 21957
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 2810 . . 3  |-  A  e. 
_V
3 shincl.2 . . . 4  |-  B  e.  SH
43elexi 2810 . . 3  |-  B  e. 
_V
52, 4intpr 3911 . 2  |-  |^| { A ,  B }  =  ( A  i^i  B )
61, 3pm3.2i 441 . . . . 5  |-  ( A  e.  SH  /\  B  e.  SH )
72, 4prss 3785 . . . . 5  |-  ( ( A  e.  SH  /\  B  e.  SH )  <->  { A ,  B }  C_  SH )
86, 7mpbi 199 . . . 4  |-  { A ,  B }  C_  SH
92prnz 3758 . . . 4  |-  { A ,  B }  =/=  (/)
108, 9pm3.2i 441 . . 3  |-  ( { A ,  B }  C_  SH  /\  { A ,  B }  =/=  (/) )
1110shintcli 21924 . 2  |-  |^| { A ,  B }  e.  SH
125, 11eqeltrri 2367 1  |-  ( A  i^i  B )  e.  SH
Colors of variables: wff set class
Syntax hints:    /\ wa 358    e. wcel 1696    =/= wne 2459    i^i cin 3164    C_ wss 3165   (/)c0 3468   {cpr 3654   |^|cint 3878   SHcsh 21524
This theorem is referenced by:  shincl  21976  shmodsi  21984  shmodi  21985  5oalem1  22249  5oalem3  22251  5oalem5  22253  5oalem6  22254  5oai  22256  3oalem2  22258  3oalem6  22262  cdj3lem1  23030
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-hilex 21595  ax-hfvadd 21596  ax-hv0cl 21599  ax-hfvmul 21601
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-sh 21802
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