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Mirrors > Home > MPE Home > Th. List > lem1 | Structured version Visualization version GIF version |
Description: A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
lem1 | ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltm1 11827 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴) | |
2 | peano2rem 11298 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ) | |
3 | ltle 11073 | . . 3 ⊢ (((𝐴 − 1) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐴 − 1) < 𝐴 → (𝐴 − 1) ≤ 𝐴)) | |
4 | 2, 3 | mpancom 685 | . 2 ⊢ (𝐴 ∈ ℝ → ((𝐴 − 1) < 𝐴 → (𝐴 − 1) ≤ 𝐴)) |
5 | 1, 4 | mpd 15 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5073 (class class class)co 7267 ℝcr 10880 1c1 10882 < clt 11019 ≤ cle 11020 − cmin 11215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-po 5498 df-so 5499 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 |
This theorem is referenced by: lem1d 11918 |
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