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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 10646 | . 2 ⊢ 0 ∈ ℝ | |
2 | 1re 10644 | . 2 ⊢ 1 ∈ ℝ | |
3 | prssi 4757 | . 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 2113 ⊆ wss 3939 {cpr 4572 ℝcr 10539 0cc0 10540 1c1 10541 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-i2m1 10608 ax-1ne0 10609 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 |
This theorem is referenced by: fprodex01 30545 indsum 31284 indsumin 31285 circlemethnat 31916 |
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