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 4018 | . 2 ⊢ (((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅ ↔ ∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝑥 ∈ (𝐴(,)𝐵)) | |
2 | elpri 4197 | . . 3 ⊢ (𝑥 ∈ {𝐴, 𝐵} → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) | |
3 | lbioo 12206 | . . . . 5 ⊢ ¬ 𝐴 ∈ (𝐴(,)𝐵) | |
4 | eleq1 2689 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 𝐴 ∈ (𝐴(,)𝐵))) | |
5 | 3, 4 | mtbiri 317 | . . . 4 ⊢ (𝑥 = 𝐴 → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
6 | ubioo 12207 | . . . . 5 ⊢ ¬ 𝐵 ∈ (𝐴(,)𝐵) | |
7 | eleq1 2689 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 𝐵 ∈ (𝐴(,)𝐵))) | |
8 | 6, 7 | mtbiri 317 | . . . 4 ⊢ (𝑥 = 𝐵 → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
9 | 5, 8 | jaoi 394 | . . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
10 | 2, 9 | syl 17 | . 2 ⊢ (𝑥 ∈ {𝐴, 𝐵} → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
11 | 1, 10 | mprgbir 2927 | 1 ⊢ ((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 383 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ∅c0 3915 {cpr 4179 (class class class)co 6650 (,)cioo 12175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-ioo 12179 |
This theorem is referenced by: iccdifioo 39741 iccdifprioo 39742 |
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