Theorem lep1

Description: A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006)

	Ref	Expression
Assertion	lep1	⊢ ( 𝐴 ∈ ℝ → 𝐴 ≤ ( 𝐴 + 1 ) )

Proof

Step	Hyp	Ref	Expression
1		ltp1	⊢ ( 𝐴 ∈ ℝ → 𝐴 < ( 𝐴 + 1 ) )
2		peano2re	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 + 1 ) ∈ ℝ )
3		ltle	⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐴 + 1 ) ∈ ℝ ) → ( 𝐴 < ( 𝐴 + 1 ) → 𝐴 ≤ ( 𝐴 + 1 ) ) )
4	2 3	mpdan	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 < ( 𝐴 + 1 ) → 𝐴 ≤ ( 𝐴 + 1 ) ) )
5	1 4	mpd	⊢ ( 𝐴 ∈ ℝ → 𝐴 ≤ ( 𝐴 + 1 ) )