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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 4646 | . . 3 ⊢ (𝑥 ∈ {𝐴, 𝐵} → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) | |
3 | lbioo 13381 | . . . . 5 ⊢ ¬ 𝐴 ∈ (𝐴(,)𝐵) | |
4 | eleq1 2817 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 𝐴 ∈ (𝐴(,)𝐵))) | |
5 | 3, 4 | mtbiri 327 | . . . 4 ⊢ (𝑥 = 𝐴 → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
6 | ubioo 13382 | . . . . 5 ⊢ ¬ 𝐵 ∈ (𝐴(,)𝐵) | |
7 | eleq1 2817 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 𝐵 ∈ (𝐴(,)𝐵))) | |
8 | 6, 7 | mtbiri 327 | . . . 4 ⊢ (𝑥 = 𝐵 → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
9 | 5, 8 | jaoi 856 | . . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
10 | 2, 9 | syl 17 | . 2 ⊢ (𝑥 ∈ {𝐴, 𝐵} → ¬ 𝑥 ∈ (𝐴(,)𝐵)) |
11 | 1, 10 | mprgbir 3064 | 1 ⊢ ((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ∩ cin 3944 ∅c0 4318 {cpr 4626 (class class class)co 7414 (,)cioo 13350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-pre-lttri 11206 ax-pre-lttrn 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 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 5143 df-opab 5205 df-mpt 5226 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-ioo 13354 |
This theorem is referenced by: iccdifioo 44894 iccdifprioo 44895 |
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