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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 9577 | . . . 4 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
2 | 1 | simprbi 479 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ≠ ∅) |
3 | noel 3878 | . . . . . 6 ⊢ ¬ 𝐴 ∈ ∅ | |
4 | 1pi 9584 | . . . . . . . . . 10 ⊢ 1𝑜 ∈ N | |
5 | ltpiord 9588 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 1𝑜 ∈ N) → (𝐴 <N 1𝑜 ↔ 𝐴 ∈ 1𝑜)) | |
6 | 4, 5 | mpan2 703 | . . . . . . . . 9 ⊢ (𝐴 ∈ N → (𝐴 <N 1𝑜 ↔ 𝐴 ∈ 1𝑜)) |
7 | df-1o 7447 | . . . . . . . . . . 11 ⊢ 1𝑜 = suc ∅ | |
8 | 7 | eleq2i 2680 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 1𝑜 ↔ 𝐴 ∈ suc ∅) |
9 | elsucg 5709 | . . . . . . . . . 10 ⊢ (𝐴 ∈ N → (𝐴 ∈ suc ∅ ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) | |
10 | 8, 9 | syl5bb 271 | . . . . . . . . 9 ⊢ (𝐴 ∈ N → (𝐴 ∈ 1𝑜 ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) |
11 | 6, 10 | bitrd 267 | . . . . . . . 8 ⊢ (𝐴 ∈ N → (𝐴 <N 1𝑜 ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅))) |
12 | 11 | biimpa 500 | . . . . . . 7 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1𝑜) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅)) |
13 | 12 | ord 391 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1𝑜) → (¬ 𝐴 ∈ ∅ → 𝐴 = ∅)) |
14 | 3, 13 | mpi 20 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐴 <N 1𝑜) → 𝐴 = ∅) |
15 | 14 | ex 449 | . . . 4 ⊢ (𝐴 ∈ N → (𝐴 <N 1𝑜 → 𝐴 = ∅)) |
16 | 15 | necon3ad 2795 | . . 3 ⊢ (𝐴 ∈ N → (𝐴 ≠ ∅ → ¬ 𝐴 <N 1𝑜)) |
17 | 2, 16 | mpd 15 | . 2 ⊢ (𝐴 ∈ N → ¬ 𝐴 <N 1𝑜) |
18 | ltrelpi 9590 | . . . . 5 ⊢ <N ⊆ (N × N) | |
19 | 18 | brel 5090 | . . . 4 ⊢ (𝐴 <N 1𝑜 → (𝐴 ∈ N ∧ 1𝑜 ∈ N)) |
20 | 19 | simpld 474 | . . 3 ⊢ (𝐴 <N 1𝑜 → 𝐴 ∈ N) |
21 | 20 | con3i 149 | . 2 ⊢ (¬ 𝐴 ∈ N → ¬ 𝐴 <N 1𝑜) |
22 | 17, 21 | pm2.61i 175 | 1 ⊢ ¬ 𝐴 <N 1𝑜 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 class class class wbr 4583 suc csuc 5642 ωcom 6957 1𝑜c1o 7440 Ncnpi 9545 <N clti 9548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-om 6958 df-1o 7447 df-ni 9573 df-lti 9576 |
This theorem is referenced by: indpi 9608 pinq 9628 archnq 9681 |
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