| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > nlt1pi | Structured version Visualization version GIF version | ||
| Description: No positive integer is less than one. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nlt1pi | ⊢ ¬ 𝐴 <N 1o |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elni 10916 | . . . 4 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
| 2 | 1 | simprbi 496 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ≠ ∅) |
| 3 | noel 4338 | . . . . . 6 ⊢ ¬ 𝐴 ∈ ∅ | |
| 4 | 1pi 10923 | . . . . . . . . . 10 ⊢ 1o ∈ N | |
| 5 | ltpiord 10927 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → (𝐴 <N 1o ↔ 𝐴 ∈ 1o)) | |
| 6 | 4, 5 | mpan2 691 | . . . . . . . . 9 ⊢ (𝐴 ∈ N → (𝐴 <N 1o ↔ 𝐴 ∈ 1o)) |
| 7 | df-1o 8506 | . . . . . . . . . . 11 ⊢ 1o = suc ∅ | |
| 8 | 7 | eleq2i 2833 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ suc ∅) |
| 9 | elsucg 6452 | . . . . . . . . . 10 ⊢ (𝐴 ∈ N → (𝐴 ∈ suc ∅ ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) | |
| 10 | 8, 9 | bitrid 283 | . . . . . . . . 9 ⊢ (𝐴 ∈ N → (𝐴 ∈ 1o ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) |
| 11 | 6, 10 | bitrd 279 | . . . . . . . 8 ⊢ (𝐴 ∈ N → (𝐴 <N 1o ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) |
| 12 | 11 | biimpa 476 | . . . . . . 7 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅)) |
| 13 | 12 | ord 865 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → (¬ 𝐴 ∈ ∅ → 𝐴 = ∅)) |
| 14 | 3, 13 | mpi 20 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → 𝐴 = ∅) |
| 15 | 14 | ex 412 | . . . 4 ⊢ (𝐴 ∈ N → (𝐴 <N 1o → 𝐴 = ∅)) |
| 16 | 15 | necon3ad 2953 | . . 3 ⊢ (𝐴 ∈ N → (𝐴 ≠ ∅ → ¬ 𝐴 <N 1o)) |
| 17 | 2, 16 | mpd 15 | . 2 ⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1o) |
| 18 | ltrelpi 10929 | . . . . 5 ⊢ <N ⊆ (N × N) | |
| 19 | 18 | brel 5750 | . . . 4 ⊢ (𝐴 <N 1o → (𝐴 ∈ N ∧ 1o ∈ N)) |
| 20 | 19 | simpld 494 | . . 3 ⊢ (𝐴 <N 1o → 𝐴 ∈ N) |
| 21 | 20 | con3i 154 | . 2 ⊢ (¬ 𝐴 ∈ N → ¬ 𝐴 <N 1o) |
| 22 | 17, 21 | pm2.61i 182 | 1 ⊢ ¬ 𝐴 <N 1o |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 class class class wbr 5143 suc csuc 6386 ωcom 7887 1oc1o 8499 Ncnpi 10884 <N clti 10887 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-om 7888 df-1o 8506 df-ni 10912 df-lti 10915 |
| This theorem is referenced by: indpi 10947 pinq 10967 archnq 11020 |
| Copyright terms: Public domain | W3C validator |