Theorem sn-ltp1

Description: ltp1 without ax-mulcom . (Contributed by SN, 13-Feb-2024)

	Ref	Expression
Assertion	sn-ltp1	⊢ ( 𝐴 ∈ ℝ → 𝐴 < ( 𝐴 + 1 ) )

Proof

Step	Hyp	Ref	Expression
1		1re	⊢ 1 ∈ ℝ
2		sn-0lt1	⊢ 0 < 1
3		sn-ltaddpos	⊢ ( ( 1 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 < 1 ↔ 𝐴 < ( 𝐴 + 1 ) ) )
4	2 3	mpbii	⊢ ( ( 1 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → 𝐴 < ( 𝐴 + 1 ) )
5	1 4	mpan	⊢ ( 𝐴 ∈ ℝ → 𝐴 < ( 𝐴 + 1 ) )