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 10700 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → -∞ ∈ ℝ*) |
3 | icomnfinre.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | 3 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝐴 ∈ ℝ*) |
5 | elinel2 4175 | . . . . 5 ⊢ (𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ) → 𝑥 ∈ ℝ) | |
6 | 5 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ ℝ) |
7 | 6 | mnfltd 12522 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → -∞ < 𝑥) |
8 | elinel1 4174 | . . . . . 6 ⊢ (𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ) → 𝑥 ∈ (-∞[,)𝐴)) | |
9 | 8 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ (-∞[,)𝐴)) |
10 | 2, 4, 9 | icoltubd 41828 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 < 𝐴) |
11 | 2, 4, 6, 7, 10 | eliood 41780 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ (-∞(,)𝐴)) |
12 | 11 | ssd 41351 | . 2 ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) ⊆ (-∞(,)𝐴)) |
13 | ioossico 12829 | . . . 4 ⊢ (-∞(,)𝐴) ⊆ (-∞[,)𝐴) | |
14 | ioossre 12801 | . . . 4 ⊢ (-∞(,)𝐴) ⊆ ℝ | |
15 | 13, 14 | ssini 4210 | . . 3 ⊢ (-∞(,)𝐴) ⊆ ((-∞[,)𝐴) ∩ ℝ) |
16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → (-∞(,)𝐴) ⊆ ((-∞[,)𝐴) ∩ ℝ)) |
17 | 12, 16 | eqssd 3986 | 1 ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 (class class class)co 7158 ℝcr 10538 -∞cmnf 10675 ℝ*cxr 10676 (,)cioo 12741 [,)cico 12743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-ioo 12745 df-ico 12747 |
This theorem is referenced by: preimaioomnf 43004 |
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