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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 10913 | . . . 4 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
2 | 1 | simprbi 496 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ≠ ∅) |
3 | noel 4343 | . . . . . 6 ⊢ ¬ 𝐴 ∈ ∅ | |
4 | 1pi 10920 | . . . . . . . . . 10 ⊢ 1o ∈ N | |
5 | ltpiord 10924 | . . . . . . . . . 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 8504 | . . . . . . . . . . 11 ⊢ 1o = suc ∅ | |
8 | 7 | eleq2i 2830 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ suc ∅) |
9 | elsucg 6453 | . . . . . . . . . 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 864 | . . . . . 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 2950 | . . 3 ⊢ (𝐴 ∈ N → (𝐴 ≠ ∅ → ¬ 𝐴 <N 1o)) |
17 | 2, 16 | mpd 15 | . 2 ⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1o) |
18 | ltrelpi 10926 | . . . . 5 ⊢ <N ⊆ (N × N) | |
19 | 18 | brel 5753 | . . . 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 847 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∅c0 4338 class class class wbr 5147 suc csuc 6387 ωcom 7886 1oc1o 8497 Ncnpi 10881 <N clti 10884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-om 7887 df-1o 8504 df-ni 10909 df-lti 10912 |
This theorem is referenced by: indpi 10944 pinq 10964 archnq 11017 |
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