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Mirrors > Home > HSE Home > Th. List > chincli | Structured version Visualization version GIF version |
Description: Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ch0le.1 | ⊢ 𝐴 ∈ Cℋ |
chjcl.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
chincli | ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ch0le.1 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
2 | 1 | elexi 3186 | . . 3 ⊢ 𝐴 ∈ V |
3 | chjcl.2 | . . . 4 ⊢ 𝐵 ∈ Cℋ | |
4 | 3 | elexi 3186 | . . 3 ⊢ 𝐵 ∈ V |
5 | 2, 4 | intpr 4445 | . 2 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
6 | 1, 3 | pm3.2i 470 | . . . . 5 ⊢ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) |
7 | 2, 4 | prss 4291 | . . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ↔ {𝐴, 𝐵} ⊆ Cℋ ) |
8 | 6, 7 | mpbi 219 | . . . 4 ⊢ {𝐴, 𝐵} ⊆ Cℋ |
9 | 2 | prnz 4253 | . . . 4 ⊢ {𝐴, 𝐵} ≠ ∅ |
10 | 8, 9 | pm3.2i 470 | . . 3 ⊢ ({𝐴, 𝐵} ⊆ Cℋ ∧ {𝐴, 𝐵} ≠ ∅) |
11 | 10 | chintcli 27574 | . 2 ⊢ ∩ {𝐴, 𝐵} ∈ Cℋ |
12 | 5, 11 | eqeltrri 2685 | 1 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∈ wcel 1977 ≠ wne 2780 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {cpr 4127 ∩ cint 4410 Cℋ cch 27170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 ax-hilex 27240 ax-hfvadd 27241 ax-hv0cl 27244 ax-hfvmul 27246 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-map 7746 df-nn 10898 df-sh 27448 df-ch 27462 |
This theorem is referenced by: chdmm1i 27720 chdmj1i 27724 chincl 27742 ledii 27779 lejdii 27781 lejdiri 27782 pjoml2i 27828 pjoml3i 27829 pjoml4i 27830 pjoml6i 27832 cmcmlem 27834 cmcm2i 27836 cmbr2i 27839 cmbr3i 27843 cmm1i 27849 fh3i 27866 fh4i 27867 cm2mi 27869 qlaxr3i 27879 osumcori 27886 osumcor2i 27887 spansnm0i 27893 5oai 27904 3oalem5 27909 3oalem6 27910 3oai 27911 pjssmii 27924 pjssge0ii 27925 pjcji 27927 pjocini 27941 mayetes3i 27972 pjssdif2i 28417 pjssdif1i 28418 pjin1i 28435 pjin3i 28437 pjclem1 28438 pjclem4 28442 pjci 28443 pjcmul1i 28444 pjcmul2i 28445 pj3si 28450 pj3cor1i 28452 stji1i 28485 stm1i 28486 stm1add3i 28490 jpi 28513 golem1 28514 golem2 28515 goeqi 28516 stcltrlem2 28520 mdslle1i 28560 mdslj1i 28562 mdslj2i 28563 mdsl1i 28564 mdsl2i 28565 mdsl2bi 28566 cvmdi 28567 mdslmd1lem1 28568 mdslmd1lem2 28569 mdslmd1i 28572 mdsldmd1i 28574 mdslmd3i 28575 mdslmd4i 28576 csmdsymi 28577 mdexchi 28578 hatomistici 28605 chrelat2i 28608 cvexchlem 28611 cvexchi 28612 sumdmdlem2 28662 mdcompli 28672 dmdcompli 28673 mddmdin0i 28674 |
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