| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ren0 | Structured version Visualization version GIF version | ||
| Description: The set of reals is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| Ref | Expression |
|---|---|
| ren0 | ⊢ ℝ ≠ ∅ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0re 11263 | . 2 ⊢ 0 ∈ ℝ | |
| 2 | 1 | ne0ii 4344 | 1 ⊢ ℝ ≠ ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ≠ wne 2940 ∅c0 4333 ℝcr 11154 0cc0 11155 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rex 3071 df-dif 3954 df-nul 4334 |
| This theorem is referenced by: limsup0 45709 limsuppnfdlem 45716 limsup10ex 45788 liminf10ex 45789 |
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