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Theorem nlt1pi 9940
 Description: No positive integer is less than one. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
Assertion
Ref Expression
nlt1pi ¬ 𝐴 <N 1𝑜

Proof of Theorem nlt1pi
StepHypRef Expression
1 elni 9910 . . . 4 (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
21simprbi 483 . . 3 (𝐴N𝐴 ≠ ∅)
3 noel 4062 . . . . . 6 ¬ 𝐴 ∈ ∅
4 1pi 9917 . . . . . . . . . 10 1𝑜N
5 ltpiord 9921 . . . . . . . . . 10 ((𝐴N ∧ 1𝑜N) → (𝐴 <N 1𝑜𝐴 ∈ 1𝑜))
64, 5mpan2 709 . . . . . . . . 9 (𝐴N → (𝐴 <N 1𝑜𝐴 ∈ 1𝑜))
7 df-1o 7730 . . . . . . . . . . 11 1𝑜 = suc ∅
87eleq2i 2831 . . . . . . . . . 10 (𝐴 ∈ 1𝑜𝐴 ∈ suc ∅)
9 elsucg 5953 . . . . . . . . . 10 (𝐴N → (𝐴 ∈ suc ∅ ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅)))
108, 9syl5bb 272 . . . . . . . . 9 (𝐴N → (𝐴 ∈ 1𝑜 ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅)))
116, 10bitrd 268 . . . . . . . 8 (𝐴N → (𝐴 <N 1𝑜 ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅)))
1211biimpa 502 . . . . . . 7 ((𝐴N𝐴 <N 1𝑜) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅))
1312ord 391 . . . . . 6 ((𝐴N𝐴 <N 1𝑜) → (¬ 𝐴 ∈ ∅ → 𝐴 = ∅))
143, 13mpi 20 . . . . 5 ((𝐴N𝐴 <N 1𝑜) → 𝐴 = ∅)
1514ex 449 . . . 4 (𝐴N → (𝐴 <N 1𝑜𝐴 = ∅))
1615necon3ad 2945 . . 3 (𝐴N → (𝐴 ≠ ∅ → ¬ 𝐴 <N 1𝑜))
172, 16mpd 15 . 2 (𝐴N → ¬ 𝐴 <N 1𝑜)
18 ltrelpi 9923 . . . . 5 <N ⊆ (N × N)
1918brel 5325 . . . 4 (𝐴 <N 1𝑜 → (𝐴N ∧ 1𝑜N))
2019simpld 477 . . 3 (𝐴 <N 1𝑜𝐴N)
2120con3i 150 . 2 𝐴N → ¬ 𝐴 <N 1𝑜)
2217, 21pm2.61i 176 1 ¬ 𝐴 <N 1𝑜
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 382   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  ∅c0 4058   class class class wbr 4804  suc csuc 5886  ωcom 7231  1𝑜c1o 7723  Ncnpi 9878
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