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Mirrors > Home > ILE Home > Th. List > ltp1 | GIF version |
Description: A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
Ref | Expression |
---|---|
ltp1 | ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 7733 | . 2 ⊢ 1 ∈ ℝ | |
2 | 0lt1 7857 | . . 3 ⊢ 0 < 1 | |
3 | ltaddpos 8182 | . . 3 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 1 ↔ 𝐴 < (𝐴 + 1))) | |
4 | 2, 3 | mpbii 147 | . 2 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → 𝐴 < (𝐴 + 1)) |
5 | 1, 4 | mpan 420 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1465 class class class wbr 3899 (class class class)co 5742 ℝcr 7587 0cc0 7588 1c1 7589 + caddc 7591 < clt 7768 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-iota 5058 df-fv 5101 df-ov 5745 df-pnf 7770 df-mnf 7771 df-ltxr 7773 |
This theorem is referenced by: lep1 8571 letrp1 8574 recp1lt1 8625 ledivp1 8629 ltp1i 8631 ltp1d 8656 uzind 9130 ge0p1rp 9441 qbtwnxr 10003 |
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