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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iooinlbub | Structured version Visualization version GIF version |
Description: An open interval has empty intersection with its bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
iooinlbub | ⊢ ((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjr 4445 | . 2 ⊢ (((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅ ↔ ∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝑥 ∈ (𝐴(,)𝐵)) | |
2 | elpri 4647 | . . 3 ⊢ (𝑥 ∈ {𝐴, 𝐵} → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) | |
3 | lbioo 13387 | . . . . 5 ⊢ ¬ 𝐴 ∈ (𝐴(,)𝐵) | |
4 | eleq1 2813 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 𝐴 ∈ (𝐴(,)𝐵))) | |
5 | 3, 4 | mtbiri 326 | . . . 4 ⊢ (𝑥 = 𝐴 → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
6 | ubioo 13388 | . . . . 5 ⊢ ¬ 𝐵 ∈ (𝐴(,)𝐵) | |
7 | eleq1 2813 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 𝐵 ∈ (𝐴(,)𝐵))) | |
8 | 6, 7 | mtbiri 326 | . . . 4 ⊢ (𝑥 = 𝐵 → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
9 | 5, 8 | jaoi 855 | . . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
10 | 2, 9 | syl 17 | . 2 ⊢ (𝑥 ∈ {𝐴, 𝐵} → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
11 | 1, 10 | mprgbir 3058 | 1 ⊢ ((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ∩ cin 3938 ∅c0 4318 {cpr 4626 (class class class)co 7416 (,)cioo 13356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-pre-lttri 11212 ax-pre-lttrn 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-ioo 13360 |
This theorem is referenced by: iccdifioo 44963 iccdifprioo 44964 |
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