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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssrexr | Structured version Visualization version GIF version |
Description: A subset of the reals is a subset of the extended reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
ssrexr.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
Ref | Expression |
---|---|
ssrexr | ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrexr.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
2 | ressxr 10674 | . 2 ⊢ ℝ ⊆ ℝ* | |
3 | 1, 2 | sstrdi 3927 | 1 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3881 ℝcr 10525 ℝ*cxr 10663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-xr 10668 |
This theorem is referenced by: limsuppnfdlem 42343 liminfval2 42410 |
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