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Theorem nlt1pi 10317
 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 10287 . . . 4 (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
21simprbi 497 . . 3 (𝐴N𝐴 ≠ ∅)
3 noel 4300 . . . . . 6 ¬ 𝐴 ∈ ∅
4 1pi 10294 . . . . . . . . . 10 1oN
5 ltpiord 10298 . . . . . . . . . 10 ((𝐴N ∧ 1oN) → (𝐴 <N 1o𝐴 ∈ 1o))
64, 5mpan2 687 . . . . . . . . 9 (𝐴N → (𝐴 <N 1o𝐴 ∈ 1o))
7 df-1o 8093 . . . . . . . . . . 11 1o = suc ∅
87eleq2i 2909 . . . . . . . . . 10 (𝐴 ∈ 1o𝐴 ∈ suc ∅)
9 elsucg 6256 . . . . . . . . . 10 (𝐴N → (𝐴 ∈ suc ∅ ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅)))
108, 9syl5bb 284 . . . . . . . . 9 (𝐴N → (𝐴 ∈ 1o ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅)))
116, 10bitrd 280 . . . . . . . 8 (𝐴N → (𝐴 <N 1o ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅)))
1211biimpa 477 . . . . . . 7 ((𝐴N𝐴 <N 1o) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅))
1312ord 860 . . . . . 6 ((𝐴N𝐴 <N 1o) → (¬ 𝐴 ∈ ∅ → 𝐴 = ∅))
143, 13mpi 20 . . . . 5 ((𝐴N𝐴 <N 1o) → 𝐴 = ∅)
1514ex 413 . . . 4 (𝐴N → (𝐴 <N 1o𝐴 = ∅))
1615necon3ad 3034 . . 3 (𝐴N → (𝐴 ≠ ∅ → ¬ 𝐴 <N 1o))
172, 16mpd 15 . 2 (𝐴N → ¬ 𝐴 <N 1o)
18 ltrelpi 10300 . . . . 5 <N ⊆ (N × N)
1918brel 5616 . . . 4 (𝐴 <N 1o → (𝐴N ∧ 1oN))
2019simpld 495 . . 3 (𝐴 <N 1o𝐴N)
2120con3i 157 . 2 𝐴N → ¬ 𝐴 <N 1o)
2217, 21pm2.61i 183 1 ¬ 𝐴 <N 1o
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∨ wo 843   = wceq 1530   ∈ wcel 2107   ≠ wne 3021  ∅c0 4295   class class class wbr 5063  suc csuc 6191  ωcom 7568  1oc1o 8086  Ncnpi 10255
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