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Mirrors > Home > MPE Home > Th. List > ordtcld3 | Structured version Visualization version GIF version |
Description: A closed interval [𝐴, 𝐵] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
ordttopon.3 | ⊢ 𝑋 = dom 𝑅 |
Ref | Expression |
---|---|
ordtcld3 | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ (𝐴𝑅𝑥 ∧ 𝑥𝑅𝐵)} ∈ (Clsd‘(ordTop‘𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inrab 3858 | . 2 ⊢ ({𝑥 ∈ 𝑋 ∣ 𝐴𝑅𝑥} ∩ {𝑥 ∈ 𝑋 ∣ 𝑥𝑅𝐵}) = {𝑥 ∈ 𝑋 ∣ (𝐴𝑅𝑥 ∧ 𝑥𝑅𝐵)} | |
2 | ordttopon.3 | . . . . 5 ⊢ 𝑋 = dom 𝑅 | |
3 | 2 | ordtcld2 20812 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ 𝐴𝑅𝑥} ∈ (Clsd‘(ordTop‘𝑅))) |
4 | 3 | 3adant3 1074 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ 𝐴𝑅𝑥} ∈ (Clsd‘(ordTop‘𝑅))) |
5 | 2 | ordtcld1 20811 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ 𝑥𝑅𝐵} ∈ (Clsd‘(ordTop‘𝑅))) |
6 | 5 | 3adant2 1073 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ 𝑥𝑅𝐵} ∈ (Clsd‘(ordTop‘𝑅))) |
7 | incld 20657 | . . 3 ⊢ (({𝑥 ∈ 𝑋 ∣ 𝐴𝑅𝑥} ∈ (Clsd‘(ordTop‘𝑅)) ∧ {𝑥 ∈ 𝑋 ∣ 𝑥𝑅𝐵} ∈ (Clsd‘(ordTop‘𝑅))) → ({𝑥 ∈ 𝑋 ∣ 𝐴𝑅𝑥} ∩ {𝑥 ∈ 𝑋 ∣ 𝑥𝑅𝐵}) ∈ (Clsd‘(ordTop‘𝑅))) | |
8 | 4, 6, 7 | syl2anc 691 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ({𝑥 ∈ 𝑋 ∣ 𝐴𝑅𝑥} ∩ {𝑥 ∈ 𝑋 ∣ 𝑥𝑅𝐵}) ∈ (Clsd‘(ordTop‘𝑅))) |
9 | 1, 8 | syl5eqelr 2693 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ (𝐴𝑅𝑥 ∧ 𝑥𝑅𝐵)} ∈ (Clsd‘(ordTop‘𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {crab 2900 ∩ cin 3539 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 ordTopcordt 15982 Clsdccld 20630 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-iin 4458 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-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-fin 7845 df-fi 8200 df-topgen 15927 df-ordt 15984 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 |
This theorem is referenced by: iccordt 20828 ordtt1 20993 |
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