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Theorem nlt1pig 6201
 Description: No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
Assertion
Ref Expression
nlt1pig (A N → ¬ A <N 1𝑜)

Proof of Theorem nlt1pig
StepHypRef Expression
1 elni 6168 . . 3 (A N ↔ (A 𝜔 A ≠ ∅))
21simprbi 260 . 2 (A NA ≠ ∅)
3 noel 3205 . . . . 5 ¬ A
4 1pi 6175 . . . . . . . . 9 1𝑜 N
5 ltpiord 6179 . . . . . . . . 9 ((A N 1𝑜 N) → (A <N 1𝑜A 1𝑜))
64, 5mpan2 403 . . . . . . . 8 (A N → (A <N 1𝑜A 1𝑜))
7 df-1o 5916 . . . . . . . . . 10 1𝑜 = suc ∅
87eleq2i 2086 . . . . . . . . 9 (A 1𝑜A suc ∅)
9 elsucg 4090 . . . . . . . . 9 (A N → (A suc ∅ ↔ (A A = ∅)))
108, 9syl5bb 181 . . . . . . . 8 (A N → (A 1𝑜 ↔ (A A = ∅)))
116, 10bitrd 177 . . . . . . 7 (A N → (A <N 1𝑜 ↔ (A A = ∅)))
1211biimpa 280 . . . . . 6 ((A N A <N 1𝑜) → (A A = ∅))
1312ord 630 . . . . 5 ((A N A <N 1𝑜) → (¬ A ∅ → A = ∅))
143, 13mpi 15 . . . 4 ((A N A <N 1𝑜) → A = ∅)
1514ex 108 . . 3 (A N → (A <N 1𝑜A = ∅))
1615necon3ad 2225 . 2 (A N → (A ≠ ∅ → ¬ A <N 1𝑜))
172, 16mpd 13 1 (A N → ¬ A <N 1𝑜)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 616   = wceq 1228   ∈ wcel 1374   ≠ wne 2186  ∅c0 3201   class class class wbr 3738  suc csuc 4051  𝜔com 4240  1𝑜c1o 5909  Ncnpi 6130
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