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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 3460 | . . 3 ⊢ 𝐴 ∈ V |
3 | chjcl.2 | . . . 4 ⊢ 𝐵 ∈ Cℋ | |
4 | 3 | elexi 3460 | . . 3 ⊢ 𝐵 ∈ V |
5 | 2, 4 | intpr 4871 | . 2 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
6 | 1, 3 | pm3.2i 474 | . . . . 5 ⊢ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) |
7 | 2, 4 | prss 4713 | . . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ↔ {𝐴, 𝐵} ⊆ Cℋ ) |
8 | 6, 7 | mpbi 233 | . . . 4 ⊢ {𝐴, 𝐵} ⊆ Cℋ |
9 | 2 | prnz 4673 | . . . 4 ⊢ {𝐴, 𝐵} ≠ ∅ |
10 | 8, 9 | pm3.2i 474 | . . 3 ⊢ ({𝐴, 𝐵} ⊆ Cℋ ∧ {𝐴, 𝐵} ≠ ∅) |
11 | 10 | chintcli 29114 | . 2 ⊢ ∩ {𝐴, 𝐵} ∈ Cℋ |
12 | 5, 11 | eqeltrri 2887 | 1 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {cpr 4527 ∩ cint 4838 Cℋ cch 28712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 ax-hilex 28782 ax-hfvadd 28783 ax-hv0cl 28786 ax-hfvmul 28788 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-map 8391 df-nn 11626 df-sh 28990 df-ch 29004 |
This theorem is referenced by: chdmm1i 29260 chdmj1i 29264 chincl 29282 ledii 29319 lejdii 29321 lejdiri 29322 pjoml2i 29368 pjoml3i 29369 pjoml4i 29370 pjoml6i 29372 cmcmlem 29374 cmcm2i 29376 cmbr2i 29379 cmbr3i 29383 cmm1i 29389 fh3i 29406 fh4i 29407 cm2mi 29409 qlaxr3i 29419 osumcori 29426 osumcor2i 29427 spansnm0i 29433 5oai 29444 3oalem5 29449 3oalem6 29450 3oai 29451 pjssmii 29464 pjssge0ii 29465 pjcji 29467 pjocini 29481 mayetes3i 29512 pjssdif2i 29957 pjssdif1i 29958 pjin1i 29975 pjin3i 29977 pjclem1 29978 pjclem4 29982 pjci 29983 pjcmul1i 29984 pjcmul2i 29985 pj3si 29990 pj3cor1i 29992 stji1i 30025 stm1i 30026 stm1add3i 30030 jpi 30053 golem1 30054 golem2 30055 goeqi 30056 stcltrlem2 30060 mdslle1i 30100 mdslj1i 30102 mdslj2i 30103 mdsl1i 30104 mdsl2i 30105 mdsl2bi 30106 cvmdi 30107 mdslmd1lem1 30108 mdslmd1lem2 30109 mdslmd1i 30112 mdsldmd1i 30114 mdslmd3i 30115 mdslmd4i 30116 csmdsymi 30117 mdexchi 30118 hatomistici 30145 chrelat2i 30148 cvexchlem 30151 cvexchi 30152 sumdmdlem2 30202 mdcompli 30212 dmdcompli 30213 mddmdin0i 30214 |
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