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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 6770 | . . 3 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
2 | 1 | simprbi 269 | . 2 ⊢ (𝐴 ∈ N → 𝐴 ≠ ∅) |
3 | noel 3273 | . . . . 5 ⊢ ¬ 𝐴 ∈ ∅ | |
4 | 1pi 6777 | . . . . . . . . 9 ⊢ 1𝑜 ∈ N | |
5 | ltpiord 6781 | . . . . . . . . 9 ⊢ ((𝐴 ∈ N ∧ 1𝑜 ∈ N) → (𝐴 <N 1𝑜 ↔ 𝐴 ∈ 1𝑜)) | |
6 | 4, 5 | mpan2 416 | . . . . . . . 8 ⊢ (𝐴 ∈ N → (𝐴 <N 1𝑜 ↔ 𝐴 ∈ 1𝑜)) |
7 | df-1o 6113 | . . . . . . . . . 10 ⊢ 1𝑜 = suc ∅ | |
8 | 7 | eleq2i 2149 | . . . . . . . . 9 ⊢ (𝐴 ∈ 1𝑜 ↔ 𝐴 ∈ suc ∅) |
9 | elsucg 4195 | . . . . . . . . 9 ⊢ (𝐴 ∈ N → (𝐴 ∈ suc ∅ ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) | |
10 | 8, 9 | syl5bb 190 | . . . . . . . 8 ⊢ (𝐴 ∈ N → (𝐴 ∈ 1𝑜 ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) |
11 | 6, 10 | bitrd 186 | . . . . . . 7 ⊢ (𝐴 ∈ N → (𝐴 <N 1𝑜 ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) |
12 | 11 | biimpa 290 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1𝑜) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅)) |
13 | 12 | ord 676 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1𝑜) → (¬ 𝐴 ∈ ∅ → 𝐴 = ∅)) |
14 | 3, 13 | mpi 15 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1𝑜) → 𝐴 = ∅) |
15 | 14 | ex 113 | . . 3 ⊢ (𝐴 ∈ N → (𝐴 <N 1𝑜 → 𝐴 = ∅)) |
16 | 15 | necon3ad 2291 | . 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 102 ↔ wb 103 ∨ wo 662 = wceq 1285 ∈ wcel 1434 ≠ wne 2249 ∅c0 3269 class class class wbr 3811 suc csuc 4156 ωcom 4368 1𝑜c1o 6106 Ncnpi 6734 <N clti 6737 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-v 2614 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-eprel 4080 df-suc 4162 df-iom 4369 df-xp 4407 df-1o 6113 df-ni 6766 df-lti 6769 |
This theorem is referenced by: caucvgsr 7250 |
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