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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 3244 | . . 3 ⊢ 𝐴 ∈ V |
3 | chjcl.2 | . . . 4 ⊢ 𝐵 ∈ Cℋ | |
4 | 3 | elexi 3244 | . . 3 ⊢ 𝐵 ∈ V |
5 | 2, 4 | intpr 4542 | . 2 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
6 | 1, 3 | pm3.2i 470 | . . . . 5 ⊢ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) |
7 | 2, 4 | prss 4383 | . . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ↔ {𝐴, 𝐵} ⊆ Cℋ ) |
8 | 6, 7 | mpbi 220 | . . . 4 ⊢ {𝐴, 𝐵} ⊆ Cℋ |
9 | 2 | prnz 4341 | . . . 4 ⊢ {𝐴, 𝐵} ≠ ∅ |
10 | 8, 9 | pm3.2i 470 | . . 3 ⊢ ({𝐴, 𝐵} ⊆ Cℋ ∧ {𝐴, 𝐵} ≠ ∅) |
11 | 10 | chintcli 28318 | . 2 ⊢ ∩ {𝐴, 𝐵} ∈ Cℋ |
12 | 5, 11 | eqeltrri 2727 | 1 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∈ wcel 2030 ≠ wne 2823 ∩ cin 3606 ⊆ wss 3607 ∅c0 3948 {cpr 4212 ∩ cint 4507 Cℋ cch 27914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-i2m1 10042 ax-1ne0 10043 ax-rrecex 10046 ax-cnre 10047 ax-hilex 27984 ax-hfvadd 27985 ax-hv0cl 27988 ax-hfvmul 27990 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-map 7901 df-nn 11059 df-sh 28192 df-ch 28206 |
This theorem is referenced by: chdmm1i 28464 chdmj1i 28468 chincl 28486 ledii 28523 lejdii 28525 lejdiri 28526 pjoml2i 28572 pjoml3i 28573 pjoml4i 28574 pjoml6i 28576 cmcmlem 28578 cmcm2i 28580 cmbr2i 28583 cmbr3i 28587 cmm1i 28593 fh3i 28610 fh4i 28611 cm2mi 28613 qlaxr3i 28623 osumcori 28630 osumcor2i 28631 spansnm0i 28637 5oai 28648 3oalem5 28653 3oalem6 28654 3oai 28655 pjssmii 28668 pjssge0ii 28669 pjcji 28671 pjocini 28685 mayetes3i 28716 pjssdif2i 29161 pjssdif1i 29162 pjin1i 29179 pjin3i 29181 pjclem1 29182 pjclem4 29186 pjci 29187 pjcmul1i 29188 pjcmul2i 29189 pj3si 29194 pj3cor1i 29196 stji1i 29229 stm1i 29230 stm1add3i 29234 jpi 29257 golem1 29258 golem2 29259 goeqi 29260 stcltrlem2 29264 mdslle1i 29304 mdslj1i 29306 mdslj2i 29307 mdsl1i 29308 mdsl2i 29309 mdsl2bi 29310 cvmdi 29311 mdslmd1lem1 29312 mdslmd1lem2 29313 mdslmd1i 29316 mdsldmd1i 29318 mdslmd3i 29319 mdslmd4i 29320 csmdsymi 29321 mdexchi 29322 hatomistici 29349 chrelat2i 29352 cvexchlem 29355 cvexchi 29356 sumdmdlem2 29406 mdcompli 29416 dmdcompli 29417 mddmdin0i 29418 |
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