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Theorem 1lt2pi 6200
 Description: One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
Assertion
Ref Expression
1lt2pi 1𝑜 <N (1𝑜 +N 1𝑜)

Proof of Theorem 1lt2pi
StepHypRef Expression
1 1onn 6004 . . . . 5 1𝑜 𝜔
2 nna0 5968 . . . . 5 (1𝑜 𝜔 → (1𝑜 +𝑜 ∅) = 1𝑜)
31, 2ax-mp 7 . . . 4 (1𝑜 +𝑜 ∅) = 1𝑜
4 0lt1o 5938 . . . . 5 1𝑜
5 peano1 4244 . . . . . 6 𝜔
6 nnaord 5993 . . . . . 6 ((∅ 𝜔 1𝑜 𝜔 1𝑜 𝜔) → (∅ 1𝑜 ↔ (1𝑜 +𝑜 ∅) (1𝑜 +𝑜 1𝑜)))
75, 1, 1, 6mp3an 1217 . . . . 5 (∅ 1𝑜 ↔ (1𝑜 +𝑜 ∅) (1𝑜 +𝑜 1𝑜))
84, 7mpbi 133 . . . 4 (1𝑜 +𝑜 ∅) (1𝑜 +𝑜 1𝑜)
93, 8eqeltrri 2093 . . 3 1𝑜 (1𝑜 +𝑜 1𝑜)
10 1pi 6175 . . . 4 1𝑜 N
11 addpiord 6176 . . . 4 ((1𝑜 N 1𝑜 N) → (1𝑜 +N 1𝑜) = (1𝑜 +𝑜 1𝑜))
1210, 10, 11mp2an 404 . . 3 (1𝑜 +N 1𝑜) = (1𝑜 +𝑜 1𝑜)
139, 12eleqtrri 2095 . 2 1𝑜 (1𝑜 +N 1𝑜)
14 addclpi 6187 . . . 4 ((1𝑜 N 1𝑜 N) → (1𝑜 +N 1𝑜) N)
1510, 10, 14mp2an 404 . . 3 (1𝑜 +N 1𝑜) N
16 ltpiord 6179 . . 3 ((1𝑜 N (1𝑜 +N 1𝑜) N) → (1𝑜 <N (1𝑜 +N 1𝑜) ↔ 1𝑜 (1𝑜 +N 1𝑜)))
1710, 15, 16mp2an 404 . 2 (1𝑜 <N (1𝑜 +N 1𝑜) ↔ 1𝑜 (1𝑜 +N 1𝑜))
1813, 17mpbir 134 1 1𝑜 <N (1𝑜 +N 1𝑜)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1228   ∈ wcel 1374  ∅c0 3201   class class class wbr 3738  𝜔com 4240  (class class class)co 5436  1𝑜c1o 5909   +𝑜 coa 5913  Ncnpi 6130   +N cpli 6131
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