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 10835 | . 2 ⊢ 0 ∈ ℝ | |
2 | 1 | ne0ii 4252 | 1 ⊢ ℝ ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2940 ∅c0 4237 ℝcr 10728 0cc0 10729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-1cn 10787 ax-addrcl 10790 ax-rnegex 10800 ax-cnre 10802 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-dif 3869 df-nul 4238 |
This theorem is referenced by: limsup0 42910 limsuppnfdlem 42917 limsup10ex 42989 liminf10ex 42990 |
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