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Theorem shincli 28561
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 3365 . . 3 𝐴 ∈ V
3 shincl.2 . . . 4 𝐵S
43elexi 3365 . . 3 𝐵 ∈ V
52, 4intpr 4644 . 2 {𝐴, 𝐵} = (𝐴𝐵)
61, 3pm3.2i 447 . . . . 5 (𝐴S𝐵S )
72, 4prss 4486 . . . . 5 ((𝐴S𝐵S ) ↔ {𝐴, 𝐵} ⊆ S )
86, 7mpbi 220 . . . 4 {𝐴, 𝐵} ⊆ S
92prnz 4445 . . . 4 {𝐴, 𝐵} ≠ ∅
108, 9pm3.2i 447 . . 3 ({𝐴, 𝐵} ⊆ S ∧ {𝐴, 𝐵} ≠ ∅)
1110shintcli 28528 . 2 {𝐴, 𝐵} ∈ S
125, 11eqeltrri 2847 1 (𝐴𝐵) ∈ S
Colors of variables: wff setvar class
Syntax hints:  wa 382  wcel 2145  wne 2943  cin 3722  wss 3723  c0 4063  {cpr 4318   cint 4611   S csh 28125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-hilex 28196  ax-hfvadd 28197  ax-hv0cl 28200  ax-hfvmul 28202
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-sh 28404
This theorem is referenced by:  shincl  28580  shmodsi  28588  shmodi  28589  5oalem1  28853  5oalem3  28855  5oalem5  28857  5oalem6  28858  5oai  28860  3oalem2  28862  3oalem6  28866  cdj3lem1  29633
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