HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  shincli Structured version   Visualization version   GIF version

Theorem shincli 29053
Description: Closure of intersection of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
shincl.1 𝐴S
shincl.2 𝐵S
Assertion
Ref Expression
shincli (𝐴𝐵) ∈ S

Proof of Theorem shincli
StepHypRef Expression
1 shincl.1 . . . 4 𝐴S
21elexi 3519 . . 3 𝐴 ∈ V
3 shincl.2 . . . 4 𝐵S
43elexi 3519 . . 3 𝐵 ∈ V
52, 4intpr 4907 . 2 {𝐴, 𝐵} = (𝐴𝐵)
61, 3pm3.2i 471 . . . . 5 (𝐴S𝐵S )
72, 4prss 4752 . . . . 5 ((𝐴S𝐵S ) ↔ {𝐴, 𝐵} ⊆ S )
86, 7mpbi 231 . . . 4 {𝐴, 𝐵} ⊆ S
92prnz 4711 . . . 4 {𝐴, 𝐵} ≠ ∅
108, 9pm3.2i 471 . . 3 ({𝐴, 𝐵} ⊆ S ∧ {𝐴, 𝐵} ≠ ∅)
1110shintcli 29020 . 2 {𝐴, 𝐵} ∈ S
125, 11eqeltrri 2915 1 (𝐴𝐵) ∈ S
Colors of variables: wff setvar class
Syntax hints:  wa 396  wcel 2107  wne 3021  cin 3939  wss 3940  c0 4295  {cpr 4566   cint 4874   S csh 28619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-hilex 28690  ax-hfvadd 28691  ax-hv0cl 28694  ax-hfvmul 28696
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-ov 7151  df-sh 28898
This theorem is referenced by:  shincl  29072  shmodsi  29080  shmodi  29081  5oalem1  29345  5oalem3  29347  5oalem5  29349  5oalem6  29350  5oai  29352  3oalem2  29354  3oalem6  29358  cdj3lem1  30125
  Copyright terms: Public domain W3C validator