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 4364 | . 2 ⊢ (((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅ ↔ ∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝑥 ∈ (𝐴(,)𝐵)) | |
2 | elpri 4563 | . . 3 ⊢ (𝑥 ∈ {𝐴, 𝐵} → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) | |
3 | lbioo 12966 | . . . . 5 ⊢ ¬ 𝐴 ∈ (𝐴(,)𝐵) | |
4 | eleq1 2825 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 𝐴 ∈ (𝐴(,)𝐵))) | |
5 | 3, 4 | mtbiri 330 | . . . 4 ⊢ (𝑥 = 𝐴 → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
6 | ubioo 12967 | . . . . 5 ⊢ ¬ 𝐵 ∈ (𝐴(,)𝐵) | |
7 | eleq1 2825 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 𝐵 ∈ (𝐴(,)𝐵))) | |
8 | 6, 7 | mtbiri 330 | . . . 4 ⊢ (𝑥 = 𝐵 → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
9 | 5, 8 | jaoi 857 | . . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
10 | 2, 9 | syl 17 | . 2 ⊢ (𝑥 ∈ {𝐴, 𝐵} → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
11 | 1, 10 | mprgbir 3076 | 1 ⊢ ((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 847 = wceq 1543 ∈ wcel 2110 ∩ cin 3865 ∅c0 4237 {cpr 4543 (class class class)co 7213 (,)cioo 12935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-pre-lttri 10803 ax-pre-lttrn 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-ioo 12939 |
This theorem is referenced by: iccdifioo 42728 iccdifprioo 42729 |
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