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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pr01ssre | Structured version Visualization version GIF version |
Description: The range of the indicator function is a subset of ℝ. (Contributed by Thierry Arnoux, 14-Aug-2017.) |
Ref | Expression |
---|---|
pr01ssre | ⊢ {0, 1} ⊆ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10439 | . 2 ⊢ 0 ∈ ℝ | |
2 | 1re 10437 | . 2 ⊢ 1 ∈ ℝ | |
3 | prssi 4624 | . 2 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ) | |
4 | 1, 2, 3 | mp2an 680 | 1 ⊢ {0, 1} ⊆ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2051 ⊆ wss 3822 {cpr 4437 ℝcr 10332 0cc0 10333 1c1 10334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-i2m1 10401 ax-1ne0 10402 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-iota 6149 df-fv 6193 df-ov 6977 |
This theorem is referenced by: fprodex01 30311 indsum 30956 indsumin 30957 circlemethnat 31592 |
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