Theorem nlim3

Description: 3 is not a limit ordinal. (Contributed by RP, 13-Dec-2024)

	Ref	Expression
Assertion	nlim3	⊢ ¬ Lim 3o

Proof

Step	Hyp	Ref	Expression
1		2on	⊢ 2o ∈ On
2		nlimsuc	⊢ ( 2o ∈ On → ¬ Lim suc 2o )
3		df-3o	⊢ 3o = suc 2o
4		limeq	⊢ ( 3o = suc 2o → ( Lim 3o ↔ Lim suc 2o ) )
5	3 4	ax-mp	⊢ ( Lim 3o ↔ Lim suc 2o )
6	2 5	sylnibr	⊢ ( 2o ∈ On → ¬ Lim 3o )
7	1 6	ax-mp	⊢ ¬ Lim 3o