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Theorem shincli 28551
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 3353 . . 3 𝐴 ∈ V
3 shincl.2 . . . 4 𝐵S
43elexi 3353 . . 3 𝐵 ∈ V
52, 4intpr 4662 . 2 {𝐴, 𝐵} = (𝐴𝐵)
61, 3pm3.2i 470 . . . . 5 (𝐴S𝐵S )
72, 4prss 4496 . . . . 5 ((𝐴S𝐵S ) ↔ {𝐴, 𝐵} ⊆ S )
86, 7mpbi 220 . . . 4 {𝐴, 𝐵} ⊆ S
92prnz 4453 . . . 4 {𝐴, 𝐵} ≠ ∅
108, 9pm3.2i 470 . . 3 ({𝐴, 𝐵} ⊆ S ∧ {𝐴, 𝐵} ≠ ∅)
1110shintcli 28518 . 2 {𝐴, 𝐵} ∈ S
125, 11eqeltrri 2836 1 (𝐴𝐵) ∈ S
Colors of variables: wff setvar class
Syntax hints:  wa 383  wcel 2139  wne 2932  cin 3714  wss 3715  c0 4058  {cpr 4323   cint 4627   S csh 28115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-hilex 28186  ax-hfvadd 28187  ax-hv0cl 28190  ax-hfvmul 28192
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-sh 28394
This theorem is referenced by:  shincl  28570  shmodsi  28578  shmodi  28579  5oalem1  28843  5oalem3  28845  5oalem5  28847  5oalem6  28848  5oai  28850  3oalem2  28852  3oalem6  28856  cdj3lem1  29623
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