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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 10637 | . 2 ⊢ 0 ∈ ℝ | |
2 | 1re 10635 | . 2 ⊢ 1 ∈ ℝ | |
3 | prssi 4747 | . 2 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ {0, 1} ⊆ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 ⊆ wss 3935 {cpr 4562 ℝcr 10530 0cc0 10531 1c1 10532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-i2m1 10599 ax-1ne0 10600 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 |
This theorem is referenced by: fprodex01 30536 indsum 31275 indsumin 31276 circlemethnat 31907 |
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