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Metamath Proof Explorer

Theorem xrltled

Description: 'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle . (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	xrltled.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ^* )
	xrltled.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ^* )
	xrltled.altb	⊢ ( 𝜑 → 𝐴 < 𝐵 )
Assertion	xrltled	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		xrltled.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ^* )
2		xrltled.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ^* )
3		xrltled.altb	⊢ ( 𝜑 → 𝐴 < 𝐵 )
4		xrltle	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) → ( 𝐴 < 𝐵 → 𝐴 ≤ 𝐵 ) )
5	1 2 4	syl2anc	⊢ ( 𝜑 → ( 𝐴 < 𝐵 → 𝐴 ≤ 𝐵 ) )
6	3 5	mpd	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )