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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 10286 | . . . 4 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
2 | 1 | simprbi 497 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ≠ ∅) |
3 | noel 4293 | . . . . . 6 ⊢ ¬ 𝐴 ∈ ∅ | |
4 | 1pi 10293 | . . . . . . . . . 10 ⊢ 1o ∈ N | |
5 | ltpiord 10297 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → (𝐴 <N 1o ↔ 𝐴 ∈ 1o)) | |
6 | 4, 5 | mpan2 687 | . . . . . . . . 9 ⊢ (𝐴 ∈ N → (𝐴 <N 1o ↔ 𝐴 ∈ 1o)) |
7 | df-1o 8091 | . . . . . . . . . . 11 ⊢ 1o = suc ∅ | |
8 | 7 | eleq2i 2901 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ suc ∅) |
9 | elsucg 6251 | . . . . . . . . . 10 ⊢ (𝐴 ∈ N → (𝐴 ∈ suc ∅ ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) | |
10 | 8, 9 | syl5bb 284 | . . . . . . . . 9 ⊢ (𝐴 ∈ N → (𝐴 ∈ 1o ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) |
11 | 6, 10 | bitrd 280 | . . . . . . . 8 ⊢ (𝐴 ∈ N → (𝐴 <N 1o ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) |
12 | 11 | biimpa 477 | . . . . . . 7 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅)) |
13 | 12 | ord 858 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → (¬ 𝐴 ∈ ∅ → 𝐴 = ∅)) |
14 | 3, 13 | mpi 20 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → 𝐴 = ∅) |
15 | 14 | ex 413 | . . . 4 ⊢ (𝐴 ∈ N → (𝐴 <N 1o → 𝐴 = ∅)) |
16 | 15 | necon3ad 3026 | . . 3 ⊢ (𝐴 ∈ N → (𝐴 ≠ ∅ → ¬ 𝐴 <N 1o)) |
17 | 2, 16 | mpd 15 | . 2 ⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1o) |
18 | ltrelpi 10299 | . . . . 5 ⊢ <N ⊆ (N × N) | |
19 | 18 | brel 5610 | . . . 4 ⊢ (𝐴 <N 1o → (𝐴 ∈ N ∧ 1o ∈ N)) |
20 | 19 | simpld 495 | . . 3 ⊢ (𝐴 <N 1o → 𝐴 ∈ N) |
21 | 20 | con3i 157 | . 2 ⊢ (¬ 𝐴 ∈ N → ¬ 𝐴 <N 1o) |
22 | 17, 21 | pm2.61i 183 | 1 ⊢ ¬ 𝐴 <N 1o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 class class class wbr 5057 suc csuc 6186 ωcom 7569 1oc1o 8084 Ncnpi 10254 <N clti 10257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-om 7570 df-1o 8091 df-ni 10282 df-lti 10285 |
This theorem is referenced by: indpi 10317 pinq 10337 archnq 10390 |
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