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Theorem 1lt2pi 10320
 Description: One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
Assertion
Ref Expression
1lt2pi 1o <N (1o +N 1o)

Proof of Theorem 1lt2pi
StepHypRef Expression
1 1onn 8252 . . . . 5 1o ∈ ω
2 nna0 8217 . . . . 5 (1o ∈ ω → (1o +o ∅) = 1o)
31, 2ax-mp 5 . . . 4 (1o +o ∅) = 1o
4 0lt1o 8116 . . . . 5 ∅ ∈ 1o
5 peano1 7585 . . . . . 6 ∅ ∈ ω
6 nnaord 8232 . . . . . 6 ((∅ ∈ ω ∧ 1o ∈ ω ∧ 1o ∈ ω) → (∅ ∈ 1o ↔ (1o +o ∅) ∈ (1o +o 1o)))
75, 1, 1, 6mp3an 1458 . . . . 5 (∅ ∈ 1o ↔ (1o +o ∅) ∈ (1o +o 1o))
84, 7mpbi 233 . . . 4 (1o +o ∅) ∈ (1o +o 1o)
93, 8eqeltrri 2890 . . 3 1o ∈ (1o +o 1o)
10 1pi 10298 . . . 4 1oN
11 addpiord 10299 . . . 4 ((1oN ∧ 1oN) → (1o +N 1o) = (1o +o 1o))
1210, 10, 11mp2an 691 . . 3 (1o +N 1o) = (1o +o 1o)
139, 12eleqtrri 2892 . 2 1o ∈ (1o +N 1o)
14 addclpi 10307 . . . 4 ((1oN ∧ 1oN) → (1o +N 1o) ∈ N)
1510, 10, 14mp2an 691 . . 3 (1o +N 1o) ∈ N
16 ltpiord 10302 . . 3 ((1oN ∧ (1o +N 1o) ∈ N) → (1o <N (1o +N 1o) ↔ 1o ∈ (1o +N 1o)))
1710, 15, 16mp2an 691 . 2 (1o <N (1o +N 1o) ↔ 1o ∈ (1o +N 1o))
1813, 17mpbir 234 1 1o <N (1o +N 1o)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2112  ∅c0 4246   class class class wbr 5033  (class class class)co 7139  ωcom 7564  1oc1o 8082   +o coa 8086  Ncnpi 10259   +N cpli 10260
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