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 9975 | . . . . 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 3762 | . . . . 5 ⊢ (𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ) → 𝑥 ∈ ℝ) | |
6 | 5 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ ℝ) |
7 | 6 | mnfltd 11834 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → -∞ < 𝑥) |
8 | elinel1 3761 | . . . . . 6 ⊢ (𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ) → 𝑥 ∈ (-∞[,)𝐴)) | |
9 | 8 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ (-∞[,)𝐴)) |
10 | 2, 4, 9 | icoltubd 38619 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 < 𝐴) |
11 | 2, 4, 6, 7, 10 | eliood 38567 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞[,)𝐴) ∩ ℝ)) → 𝑥 ∈ (-∞(,)𝐴)) |
12 | 11 | ssd 38278 | . 2 ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) ⊆ (-∞(,)𝐴)) |
13 | ioossico 12133 | . . . 4 ⊢ (-∞(,)𝐴) ⊆ (-∞[,)𝐴) | |
14 | ioossre 12106 | . . . 4 ⊢ (-∞(,)𝐴) ⊆ ℝ | |
15 | 13, 14 | ssini 3798 | . . 3 ⊢ (-∞(,)𝐴) ⊆ ((-∞[,)𝐴) ∩ ℝ) |
16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → (-∞(,)𝐴) ⊆ ((-∞[,)𝐴) ∩ ℝ)) |
17 | 12, 16 | eqssd 3585 | 1 ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 (class class class)co 6549 ℝcr 9814 -∞cmnf 9951 ℝ*cxr 9952 (,)cioo 12046 [,)cico 12048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ioo 12050 df-ico 12052 |
This theorem is referenced by: preimaioomnf 39606 |
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