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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 9919 | . 2 ⊢ 0 ∈ ℝ | |
2 | 1re 9918 | . 2 ⊢ 1 ∈ ℝ | |
3 | prssi 4293 | . 2 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ {0, 1} ⊆ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ⊆ wss 3540 {cpr 4127 ℝcr 9814 0cc0 9815 1c1 9816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: indsum 29412 |
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