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Mirrors > Home > ILE Home > Th. List > nlt1pig | GIF version |
Description: No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Ref | Expression |
---|---|
nlt1pig | ⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elni 7109 | . . 3 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
2 | 1 | simprbi 273 | . 2 ⊢ (𝐴 ∈ N → 𝐴 ≠ ∅) |
3 | noel 3362 | . . . . 5 ⊢ ¬ 𝐴 ∈ ∅ | |
4 | 1pi 7116 | . . . . . . . . 9 ⊢ 1o ∈ N | |
5 | ltpiord 7120 | . . . . . . . . 9 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → (𝐴 <N 1o ↔ 𝐴 ∈ 1o)) | |
6 | 4, 5 | mpan2 421 | . . . . . . . 8 ⊢ (𝐴 ∈ N → (𝐴 <N 1o ↔ 𝐴 ∈ 1o)) |
7 | df-1o 6306 | . . . . . . . . . 10 ⊢ 1o = suc ∅ | |
8 | 7 | eleq2i 2204 | . . . . . . . . 9 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ suc ∅) |
9 | elsucg 4321 | . . . . . . . . 9 ⊢ (𝐴 ∈ N → (𝐴 ∈ suc ∅ ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) | |
10 | 8, 9 | syl5bb 191 | . . . . . . . 8 ⊢ (𝐴 ∈ N → (𝐴 ∈ 1o ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) |
11 | 6, 10 | bitrd 187 | . . . . . . 7 ⊢ (𝐴 ∈ N → (𝐴 <N 1o ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) |
12 | 11 | biimpa 294 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅)) |
13 | 12 | ord 713 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → (¬ 𝐴 ∈ ∅ → 𝐴 = ∅)) |
14 | 3, 13 | mpi 15 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → 𝐴 = ∅) |
15 | 14 | ex 114 | . . 3 ⊢ (𝐴 ∈ N → (𝐴 <N 1o → 𝐴 = ∅)) |
16 | 15 | necon3ad 2348 | . 2 ⊢ (𝐴 ∈ N → (𝐴 ≠ ∅ → ¬ 𝐴 <N 1o)) |
17 | 2, 16 | mpd 13 | 1 ⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1o) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ≠ wne 2306 ∅c0 3358 class class class wbr 3924 suc csuc 4282 ωcom 4499 1oc1o 6299 Ncnpi 7073 <N clti 7076 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-eprel 4206 df-suc 4288 df-iom 4500 df-xp 4540 df-1o 6306 df-ni 7105 df-lti 7108 |
This theorem is referenced by: caucvgsr 7603 |
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