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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 4299 | . 2 ⊢ ({𝑥 ∈ 𝑋 ∣ 𝐴𝑅𝑥} ∩ {𝑥 ∈ 𝑋 ∣ 𝑥𝑅𝐵}) = {𝑥 ∈ 𝑋 ∣ (𝐴𝑅𝑥 ∧ 𝑥𝑅𝐵)} | |
2 | ordttopon.3 | . . . . 5 ⊢ 𝑋 = dom 𝑅 | |
3 | 2 | ordtcld2 22626 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ 𝐴𝑅𝑥} ∈ (Clsd‘(ordTop‘𝑅))) |
4 | 3 | 3adant3 1132 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ 𝐴𝑅𝑥} ∈ (Clsd‘(ordTop‘𝑅))) |
5 | 2 | ordtcld1 22625 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ 𝑥𝑅𝐵} ∈ (Clsd‘(ordTop‘𝑅))) |
6 | incld 22471 | . . 3 ⊢ (({𝑥 ∈ 𝑋 ∣ 𝐴𝑅𝑥} ∈ (Clsd‘(ordTop‘𝑅)) ∧ {𝑥 ∈ 𝑋 ∣ 𝑥𝑅𝐵} ∈ (Clsd‘(ordTop‘𝑅))) → ({𝑥 ∈ 𝑋 ∣ 𝐴𝑅𝑥} ∩ {𝑥 ∈ 𝑋 ∣ 𝑥𝑅𝐵}) ∈ (Clsd‘(ordTop‘𝑅))) | |
7 | 4, 5, 6 | 3imp3i2an 1345 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ({𝑥 ∈ 𝑋 ∣ 𝐴𝑅𝑥} ∩ {𝑥 ∈ 𝑋 ∣ 𝑥𝑅𝐵}) ∈ (Clsd‘(ordTop‘𝑅))) |
8 | 1, 7 | eqeltrrid 2837 | 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3429 ∩ cin 3940 class class class wbr 5138 dom cdm 5666 ‘cfv 6529 ordTopcordt 17424 Clsdccld 22444 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-om 7836 df-1o 8445 df-er 8683 df-en 8920 df-fin 8923 df-fi 9385 df-topgen 17368 df-ordt 17426 df-top 22320 df-topon 22337 df-bases 22373 df-cld 22447 |
This theorem is referenced by: iccordt 22642 ordtt1 22807 |
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