Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrssre | Structured version Visualization version GIF version |
Description: A subset of extended reals that does not contain +∞ and -∞ is a subset of the reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
xrssre.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
xrssre.2 | ⊢ (𝜑 → ¬ +∞ ∈ 𝐴) |
xrssre.3 | ⊢ (𝜑 → ¬ -∞ ∈ 𝐴) |
Ref | Expression |
---|---|
xrssre | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrssre.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) | |
2 | ssxr 9986 | . . . . 5 ⊢ (𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) |
4 | 3orass 1034 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (𝐴 ⊆ ℝ ∨ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))) | |
5 | 3, 4 | sylib 207 | . . 3 ⊢ (𝜑 → (𝐴 ⊆ ℝ ∨ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))) |
6 | 5 | orcomd 402 | . 2 ⊢ (𝜑 → ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ)) |
7 | xrssre.2 | . . . 4 ⊢ (𝜑 → ¬ +∞ ∈ 𝐴) | |
8 | xrssre.3 | . . . 4 ⊢ (𝜑 → ¬ -∞ ∈ 𝐴) | |
9 | 7, 8 | jca 553 | . . 3 ⊢ (𝜑 → (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴)) |
10 | ioran 510 | . . 3 ⊢ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴)) | |
11 | 9, 10 | sylibr 223 | . 2 ⊢ (𝜑 → ¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) |
12 | df-or 384 | . . 3 ⊢ (((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ) ↔ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ)) | |
13 | 12 | biimpi 205 | . 2 ⊢ (((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ) → (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ)) |
14 | 6, 11, 13 | sylc 63 | 1 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 ∨ w3o 1030 ∈ wcel 1977 ⊆ wss 3540 ℝcr 9814 +∞cpnf 9950 -∞cmnf 9951 ℝ*cxr 9952 |
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-resscn 9872 |
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-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 |
This theorem is referenced by: (None) |
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