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| Mirrors > Home > MPE Home > Th. List > Mathboxes > icomnfinre | Structured version Visualization version GIF version | ||
| Description: A left-closed, right-open, interval of extended reals, intersected with the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
| Ref | Expression |
|---|---|
| icomnfinre.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
| Ref | Expression |
|---|---|
| icomnfinre | ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mnfxr 11318 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
| 2 | 1 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → -∞ ∈ ℝ*) |
| 3 | icomnfinre.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
| 4 | 3 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝐴 ∈ ℝ*) |
| 5 | elinel2 4202 | . . . . 5 ⊢ (𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ) → 𝑥 ∈ ℝ) | |
| 6 | 5 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ ℝ) |
| 7 | 6 | mnfltd 13166 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → -∞ < 𝑥) |
| 8 | elinel1 4201 | . . . . . 6 ⊢ (𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ) → 𝑥 ∈ (-∞[,)𝐴)) | |
| 9 | 8 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ (-∞[,)𝐴)) |
| 10 | 2, 4, 9 | icoltubd 45558 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 < 𝐴) |
| 11 | 2, 4, 6, 7, 10 | eliood 45511 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ (-∞(,)𝐴)) |
| 12 | 11 | ssd 45085 | . 2 ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) ⊆ (-∞(,)𝐴)) |
| 13 | ioossico 13478 | . . . 4 ⊢ (-∞(,)𝐴) ⊆ (-∞[,)𝐴) | |
| 14 | ioossre 13448 | . . . 4 ⊢ (-∞(,)𝐴) ⊆ ℝ | |
| 15 | 13, 14 | ssini 4240 | . . 3 ⊢ (-∞(,)𝐴) ⊆ ((-∞[,)𝐴) ∩ ℝ) |
| 16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → (-∞(,)𝐴) ⊆ ((-∞[,)𝐴) ∩ ℝ)) |
| 17 | 12, 16 | eqssd 4001 | 1 ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 (class class class)co 7431 ℝcr 11154 -∞cmnf 11293 ℝ*cxr 11294 (,)cioo 13387 [,)cico 13389 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391 df-ico 13393 |
| This theorem is referenced by: preimaioomnf 46734 |
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