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Theorem nlt1pig 6439
 Description: No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
Assertion
Ref Expression
nlt1pig (𝐴N → ¬ 𝐴 <N 1𝑜)

Proof of Theorem nlt1pig
StepHypRef Expression
1 elni 6406 . . 3 (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
21simprbi 260 . 2 (𝐴N𝐴 ≠ ∅)
3 noel 3228 . . . . 5 ¬ 𝐴 ∈ ∅
4 1pi 6413 . . . . . . . . 9 1𝑜N
5 ltpiord 6417 . . . . . . . . 9 ((𝐴N ∧ 1𝑜N) → (𝐴 <N 1𝑜𝐴 ∈ 1𝑜))
64, 5mpan2 401 . . . . . . . 8 (𝐴N → (𝐴 <N 1𝑜𝐴 ∈ 1𝑜))
7 df-1o 6001 . . . . . . . . . 10 1𝑜 = suc ∅
87eleq2i 2104 . . . . . . . . 9 (𝐴 ∈ 1𝑜𝐴 ∈ suc ∅)
9 elsucg 4141 . . . . . . . . 9 (𝐴N → (𝐴 ∈ suc ∅ ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅)))
108, 9syl5bb 181 . . . . . . . 8 (𝐴N → (𝐴 ∈ 1𝑜 ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅)))
116, 10bitrd 177 . . . . . . 7 (𝐴N → (𝐴 <N 1𝑜 ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅)))
1211biimpa 280 . . . . . 6 ((𝐴N𝐴 <N 1𝑜) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅))
1312ord 643 . . . . 5 ((𝐴N𝐴 <N 1𝑜) → (¬ 𝐴 ∈ ∅ → 𝐴 = ∅))
143, 13mpi 15 . . . 4 ((𝐴N𝐴 <N 1𝑜) → 𝐴 = ∅)
1514ex 108 . . 3 (𝐴N → (𝐴 <N 1𝑜𝐴 = ∅))
1615necon3ad 2247 . 2 (𝐴N → (𝐴 ≠ ∅ → ¬ 𝐴 <N 1𝑜))
172, 16mpd 13 1 (𝐴N → ¬ 𝐴 <N 1𝑜)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   = wceq 1243   ∈ wcel 1393   ≠ wne 2204  ∅c0 3224   class class class wbr 3764  suc csuc 4102  ωcom 4313  1𝑜c1o 5994  Ncnpi 6370
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