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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 3511 | . . 3 ⊢ 𝐴 ∈ V |
3 | chjcl.2 | . . . 4 ⊢ 𝐵 ∈ Cℋ | |
4 | 3 | elexi 3511 | . . 3 ⊢ 𝐵 ∈ V |
5 | 2, 4 | intpr 4900 | . 2 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
6 | 1, 3 | pm3.2i 471 | . . . . 5 ⊢ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) |
7 | 2, 4 | prss 4745 | . . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ↔ {𝐴, 𝐵} ⊆ Cℋ ) |
8 | 6, 7 | mpbi 231 | . . . 4 ⊢ {𝐴, 𝐵} ⊆ Cℋ |
9 | 2 | prnz 4704 | . . . 4 ⊢ {𝐴, 𝐵} ≠ ∅ |
10 | 8, 9 | pm3.2i 471 | . . 3 ⊢ ({𝐴, 𝐵} ⊆ Cℋ ∧ {𝐴, 𝐵} ≠ ∅) |
11 | 10 | chintcli 29035 | . 2 ⊢ ∩ {𝐴, 𝐵} ∈ Cℋ |
12 | 5, 11 | eqeltrri 2907 | 1 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∈ wcel 2105 ≠ wne 3013 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 {cpr 4559 ∩ cint 4867 Cℋ cch 28633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-1cn 10583 ax-addcl 10585 ax-hilex 28703 ax-hfvadd 28704 ax-hv0cl 28707 ax-hfvmul 28709 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-map 8397 df-nn 11627 df-sh 28911 df-ch 28925 |
This theorem is referenced by: chdmm1i 29181 chdmj1i 29185 chincl 29203 ledii 29240 lejdii 29242 lejdiri 29243 pjoml2i 29289 pjoml3i 29290 pjoml4i 29291 pjoml6i 29293 cmcmlem 29295 cmcm2i 29297 cmbr2i 29300 cmbr3i 29304 cmm1i 29310 fh3i 29327 fh4i 29328 cm2mi 29330 qlaxr3i 29340 osumcori 29347 osumcor2i 29348 spansnm0i 29354 5oai 29365 3oalem5 29370 3oalem6 29371 3oai 29372 pjssmii 29385 pjssge0ii 29386 pjcji 29388 pjocini 29402 mayetes3i 29433 pjssdif2i 29878 pjssdif1i 29879 pjin1i 29896 pjin3i 29898 pjclem1 29899 pjclem4 29903 pjci 29904 pjcmul1i 29905 pjcmul2i 29906 pj3si 29911 pj3cor1i 29913 stji1i 29946 stm1i 29947 stm1add3i 29951 jpi 29974 golem1 29975 golem2 29976 goeqi 29977 stcltrlem2 29981 mdslle1i 30021 mdslj1i 30023 mdslj2i 30024 mdsl1i 30025 mdsl2i 30026 mdsl2bi 30027 cvmdi 30028 mdslmd1lem1 30029 mdslmd1lem2 30030 mdslmd1i 30033 mdsldmd1i 30035 mdslmd3i 30036 mdslmd4i 30037 csmdsymi 30038 mdexchi 30039 hatomistici 30066 chrelat2i 30069 cvexchlem 30072 cvexchi 30073 sumdmdlem2 30123 mdcompli 30133 dmdcompli 30134 mddmdin0i 30135 |
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