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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 11302 | . 2 ⊢ ℝ ⊆ ℝ* | |
3 | 1, 2 | sstrdi 4007 | 1 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3962 ℝcr 11151 ℝ*cxr 11291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-un 3967 df-ss 3979 df-xr 11296 |
This theorem is referenced by: limsuppnfdlem 45656 limsupvaluz2 45693 liminfval2 45723 |
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