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Theorem nlt1pig 6954
 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 6921 . . 3 (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
21simprbi 270 . 2 (𝐴N𝐴 ≠ ∅)
3 noel 3291 . . . . 5 ¬ 𝐴 ∈ ∅
4 1pi 6928 . . . . . . . . 9 1oN
5 ltpiord 6932 . . . . . . . . 9 ((𝐴N ∧ 1oN) → (𝐴 <N 1o𝐴 ∈ 1o))
64, 5mpan2 417 . . . . . . . 8 (𝐴N → (𝐴 <N 1o𝐴 ∈ 1o))
7 df-1o 6195 . . . . . . . . . 10 1o = suc ∅
87eleq2i 2155 . . . . . . . . 9 (𝐴 ∈ 1o𝐴 ∈ suc ∅)
9 elsucg 4240 . . . . . . . . 9 (𝐴N → (𝐴 ∈ suc ∅ ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅)))
108, 9syl5bb 191 . . . . . . . 8 (𝐴N → (𝐴 ∈ 1o ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅)))
116, 10bitrd 187 . . . . . . 7 (𝐴N → (𝐴 <N 1o ↔ (𝐴 ∈ ∅ ∨ 𝐴 = ∅)))
1211biimpa 291 . . . . . 6 ((𝐴N𝐴 <N 1o) → (𝐴 ∈ ∅ ∨ 𝐴 = ∅))
1312ord 679 . . . . 5 ((𝐴N𝐴 <N 1o) → (¬ 𝐴 ∈ ∅ → 𝐴 = ∅))
143, 13mpi 15 . . . 4 ((𝐴N𝐴 <N 1o) → 𝐴 = ∅)
1514ex 114 . . 3 (𝐴N → (𝐴 <N 1o𝐴 = ∅))
1615necon3ad 2298 . 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 103   ↔ wb 104   ∨ wo 665   = wceq 1290   ∈ wcel 1439   ≠ wne 2256  ∅c0 3287   class class class wbr 3851  suc csuc 4201  ωcom 4418  1oc1o 6188  Ncnpi 6885
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