Theorem ltm1

Description: A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006)

	Ref	Expression
Assertion	ltm1	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − 1 ) < 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		0lt1	⊢ 0 < 1
2		0re	⊢ 0 ∈ ℝ
3		1re	⊢ 1 ∈ ℝ
4		ltsub2	⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 < 1 ↔ ( 𝐴 − 1 ) < ( 𝐴 − 0 ) ) )
5	2 3 4	mp3an12	⊢ ( 𝐴 ∈ ℝ → ( 0 < 1 ↔ ( 𝐴 − 1 ) < ( 𝐴 − 0 ) ) )
6	1 5	mpbii	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − 1 ) < ( 𝐴 − 0 ) )
7		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
8	7	subid1d	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − 0 ) = 𝐴 )
9	6 8	breqtrd	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − 1 ) < 𝐴 )