Mathbox for Glauco Siliprandi |
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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 10096 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → -∞ ∈ ℝ*) |
3 | icomnfinre.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | 3 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝐴 ∈ ℝ*) |
5 | elinel2 3800 | . . . . 5 ⊢ (𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ) → 𝑥 ∈ ℝ) | |
6 | 5 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ ℝ) |
7 | 6 | mnfltd 11958 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → -∞ < 𝑥) |
8 | elinel1 3799 | . . . . . 6 ⊢ (𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ) → 𝑥 ∈ (-∞[,)𝐴)) | |
9 | 8 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ (-∞[,)𝐴)) |
10 | 2, 4, 9 | icoltubd 39772 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 < 𝐴) |
11 | 2, 4, 6, 7, 10 | eliood 39720 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ (-∞(,)𝐴)) |
12 | 11 | ssd 39252 | . 2 ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) ⊆ (-∞(,)𝐴)) |
13 | ioossico 12262 | . . . 4 ⊢ (-∞(,)𝐴) ⊆ (-∞[,)𝐴) | |
14 | ioossre 12235 | . . . 4 ⊢ (-∞(,)𝐴) ⊆ ℝ | |
15 | 13, 14 | ssini 3836 | . . 3 ⊢ (-∞(,)𝐴) ⊆ ((-∞[,)𝐴) ∩ ℝ) |
16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → (-∞(,)𝐴) ⊆ ((-∞[,)𝐴) ∩ ℝ)) |
17 | 12, 16 | eqssd 3620 | 1 ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 (class class class)co 6650 ℝcr 9935 -∞cmnf 10072 ℝ*cxr 10073 (,)cioo 12175 [,)cico 12177 |
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-le 10080 df-ioo 12179 df-ico 12181 |
This theorem is referenced by: preimaioomnf 40929 |
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