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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 3503 | . . 3 ⊢ 𝐴 ∈ V |
| 3 | chjcl.2 | . . . 4 ⊢ 𝐵 ∈ Cℋ | |
| 4 | 3 | elexi 3503 | . . 3 ⊢ 𝐵 ∈ V |
| 5 | 2, 4 | intpr 4982 | . 2 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
| 6 | 1, 3 | pm3.2i 470 | . . . . 5 ⊢ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) |
| 7 | 2, 4 | prss 4820 | . . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ↔ {𝐴, 𝐵} ⊆ Cℋ ) |
| 8 | 6, 7 | mpbi 230 | . . . 4 ⊢ {𝐴, 𝐵} ⊆ Cℋ |
| 9 | 2 | prnz 4777 | . . . 4 ⊢ {𝐴, 𝐵} ≠ ∅ |
| 10 | 8, 9 | pm3.2i 470 | . . 3 ⊢ ({𝐴, 𝐵} ⊆ Cℋ ∧ {𝐴, 𝐵} ≠ ∅) |
| 11 | 10 | chintcli 31350 | . 2 ⊢ ∩ {𝐴, 𝐵} ∈ Cℋ |
| 12 | 5, 11 | eqeltrri 2838 | 1 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {cpr 4628 ∩ cint 4946 Cℋ cch 30948 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 ax-hilex 31018 ax-hfvadd 31019 ax-hv0cl 31022 ax-hfvmul 31024 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-map 8868 df-nn 12267 df-sh 31226 df-ch 31240 |
| This theorem is referenced by: chdmm1i 31496 chdmj1i 31500 chincl 31518 ledii 31555 lejdii 31557 lejdiri 31558 pjoml2i 31604 pjoml3i 31605 pjoml4i 31606 pjoml6i 31608 cmcmlem 31610 cmcm2i 31612 cmbr2i 31615 cmbr3i 31619 cmm1i 31625 fh3i 31642 fh4i 31643 cm2mi 31645 qlaxr3i 31655 osumcori 31662 osumcor2i 31663 spansnm0i 31669 5oai 31680 3oalem5 31685 3oalem6 31686 3oai 31687 pjssmii 31700 pjssge0ii 31701 pjcji 31703 pjocini 31717 mayetes3i 31748 pjssdif2i 32193 pjssdif1i 32194 pjin1i 32211 pjin3i 32213 pjclem1 32214 pjclem4 32218 pjci 32219 pjcmul1i 32220 pjcmul2i 32221 pj3si 32226 pj3cor1i 32228 stji1i 32261 stm1i 32262 stm1add3i 32266 jpi 32289 golem1 32290 golem2 32291 goeqi 32292 stcltrlem2 32296 mdslle1i 32336 mdslj1i 32338 mdslj2i 32339 mdsl1i 32340 mdsl2i 32341 mdsl2bi 32342 cvmdi 32343 mdslmd1lem1 32344 mdslmd1lem2 32345 mdslmd1i 32348 mdsldmd1i 32350 mdslmd3i 32351 mdslmd4i 32352 csmdsymi 32353 mdexchi 32354 hatomistici 32381 chrelat2i 32384 cvexchlem 32387 cvexchi 32388 sumdmdlem2 32438 mdcompli 32448 dmdcompli 32449 mddmdin0i 32450 |
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