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Theorem shincli 28746
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 3401 . . 3 𝐴 ∈ V
3 shincl.2 . . . 4 𝐵S
43elexi 3401 . . 3 𝐵 ∈ V
52, 4intpr 4700 . 2 {𝐴, 𝐵} = (𝐴𝐵)
61, 3pm3.2i 463 . . . . 5 (𝐴S𝐵S )
72, 4prss 4539 . . . . 5 ((𝐴S𝐵S ) ↔ {𝐴, 𝐵} ⊆ S )
86, 7mpbi 222 . . . 4 {𝐴, 𝐵} ⊆ S
92prnz 4498 . . . 4 {𝐴, 𝐵} ≠ ∅
108, 9pm3.2i 463 . . 3 ({𝐴, 𝐵} ⊆ S ∧ {𝐴, 𝐵} ≠ ∅)
1110shintcli 28713 . 2 {𝐴, 𝐵} ∈ S
125, 11eqeltrri 2875 1 (𝐴𝐵) ∈ S
Colors of variables: wff setvar class
Syntax hints:  wa 385  wcel 2157  wne 2971  cin 3768  wss 3769  c0 4115  {cpr 4370   cint 4667   S csh 28310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-hilex 28381  ax-hfvadd 28382  ax-hv0cl 28385  ax-hfvmul 28387
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-sh 28589
This theorem is referenced by:  shincl  28765  shmodsi  28773  shmodi  28774  5oalem1  29038  5oalem3  29040  5oalem5  29042  5oalem6  29043  5oai  29045  3oalem2  29047  3oalem6  29051  cdj3lem1  29818
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