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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 10298 | . . . 4 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
2 | 1 | simprbi 499 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ≠ ∅) |
3 | noel 4296 | . . . . . 6 ⊢ ¬ 𝐴 ∈ ∅ | |
4 | 1pi 10305 | . . . . . . . . . 10 ⊢ 1o ∈ N | |
5 | ltpiord 10309 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → (𝐴 <N 1o ↔ 𝐴 ∈ 1o)) | |
6 | 4, 5 | mpan2 689 | . . . . . . . . 9 ⊢ (𝐴 ∈ N → (𝐴 <N 1o ↔ 𝐴 ∈ 1o)) |
7 | df-1o 8102 | . . . . . . . . . . 11 ⊢ 1o = suc ∅ | |
8 | 7 | eleq2i 2904 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ suc ∅) |
9 | elsucg 6258 | . . . . . . . . . 10 ⊢ (𝐴 ∈ N → (𝐴 ∈ suc ∅ ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) | |
10 | 8, 9 | syl5bb 285 | . . . . . . . . 9 ⊢ (𝐴 ∈ N → (𝐴 ∈ 1o ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) |
11 | 6, 10 | bitrd 281 | . . . . . . . 8 ⊢ (𝐴 ∈ N → (𝐴 <N 1o ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) |
12 | 11 | biimpa 479 | . . . . . . 7 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅)) |
13 | 12 | ord 860 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → (¬ 𝐴 ∈ ∅ → 𝐴 = ∅)) |
14 | 3, 13 | mpi 20 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1o) → 𝐴 = ∅) |
15 | 14 | ex 415 | . . . 4 ⊢ (𝐴 ∈ N → (𝐴 <N 1o → 𝐴 = ∅)) |
16 | 15 | necon3ad 3029 | . . 3 ⊢ (𝐴 ∈ N → (𝐴 ≠ ∅ → ¬ 𝐴 <N 1o)) |
17 | 2, 16 | mpd 15 | . 2 ⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1o) |
18 | ltrelpi 10311 | . . . . 5 ⊢ <N ⊆ (N × N) | |
19 | 18 | brel 5617 | . . . 4 ⊢ (𝐴 <N 1o → (𝐴 ∈ N ∧ 1o ∈ N)) |
20 | 19 | simpld 497 | . . 3 ⊢ (𝐴 <N 1o → 𝐴 ∈ N) |
21 | 20 | con3i 157 | . 2 ⊢ (¬ 𝐴 ∈ N → ¬ 𝐴 <N 1o) |
22 | 17, 21 | pm2.61i 184 | 1 ⊢ ¬ 𝐴 <N 1o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 class class class wbr 5066 suc csuc 6193 ωcom 7580 1oc1o 8095 Ncnpi 10266 <N clti 10269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-om 7581 df-1o 8102 df-ni 10294 df-lti 10297 |
This theorem is referenced by: indpi 10329 pinq 10349 archnq 10402 |
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