Metamath Proof Explorer


Theorem 2ellim

Description: A limit ordinal contains 2. (Contributed by BTernaryTau, 1-Dec-2024)

Ref Expression
Assertion 2ellim ( Lim 𝐴 → 2o𝐴 )

Proof

Step Hyp Ref Expression
1 nlim0 ¬ Lim ∅
2 limeq ( 𝐴 = ∅ → ( Lim 𝐴 ↔ Lim ∅ ) )
3 1 2 mtbiri ( 𝐴 = ∅ → ¬ Lim 𝐴 )
4 3 necon2ai ( Lim 𝐴𝐴 ≠ ∅ )
5 nlim1 ¬ Lim 1o
6 limeq ( 𝐴 = 1o → ( Lim 𝐴 ↔ Lim 1o ) )
7 5 6 mtbiri ( 𝐴 = 1o → ¬ Lim 𝐴 )
8 7 necon2ai ( Lim 𝐴𝐴 ≠ 1o )
9 nlim2 ¬ Lim 2o
10 limeq ( 𝐴 = 2o → ( Lim 𝐴 ↔ Lim 2o ) )
11 9 10 mtbiri ( 𝐴 = 2o → ¬ Lim 𝐴 )
12 11 necon2ai ( Lim 𝐴𝐴 ≠ 2o )
13 limord ( Lim 𝐴 → Ord 𝐴 )
14 ord2eln012 ( Ord 𝐴 → ( 2o𝐴 ↔ ( 𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o𝐴 ≠ 2o ) ) )
15 13 14 syl ( Lim 𝐴 → ( 2o𝐴 ↔ ( 𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o𝐴 ≠ 2o ) ) )
16 4 8 12 15 mpbir3and ( Lim 𝐴 → 2o𝐴 )