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Mirrors > Home > ILE Home > Th. List > ltp1i | GIF version |
Description: A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
Ref | Expression |
---|---|
ltplus1.1 | ⊢ 𝐴 ∈ ℝ |
Ref | Expression |
---|---|
ltp1i | ⊢ 𝐴 < (𝐴 + 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltplus1.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
2 | ltp1 8368 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ 𝐴 < (𝐴 + 1) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1439 class class class wbr 3853 (class class class)co 5668 ℝcr 7412 1c1 7414 + caddc 7416 < clt 7585 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-addcom 7508 ax-addass 7510 ax-i2m1 7513 ax-0lt1 7514 ax-0id 7516 ax-rnegex 7517 ax-pre-ltadd 7524 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2624 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-br 3854 df-opab 3908 df-xp 4460 df-iota 4995 df-fv 5038 df-ov 5671 df-pnf 7587 df-mnf 7588 df-ltxr 7590 |
This theorem is referenced by: 1lt2 8648 2lt3 8649 3lt4 8651 4lt5 8654 5lt6 8658 6lt7 8663 7lt8 8669 8lt9 8676 9lt10 9070 |
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