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Mirrors > Home > ILE Home > Th. List > lem1d | GIF version |
Description: A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
ltp1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
lem1d | ⊢ (𝜑 → (𝐴 − 1) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltp1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | lem1 8752 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ≤ 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 − 1) ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2141 class class class wbr 3987 (class class class)co 5851 ℝcr 7762 1c1 7764 ≤ cle 7944 − cmin 8079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-addcom 7863 ax-addass 7865 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-0id 7871 ax-rnegex 7872 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-lttrn 7877 ax-pre-ltadd 7879 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 |
This theorem is referenced by: fzossrbm1 10118 iseqf1olemqcl 10431 iseqf1olemnab 10433 iseqf1olemab 10434 seq3f1olemqsumkj 10443 seq3f1olemqsum 10445 seq3coll 10766 |
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