Theorem supxrmnf2

Description: Removing minus infinity from a set does not affect its supremum. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Assertion	supxrmnf2	⊢ ( 𝐴 ⊆ ℝ* → sup ( ( 𝐴 ∖ { -∞ } ) , ℝ* , < ) = sup ( 𝐴 , ℝ* , < ) )

Proof

Step	Hyp	Ref	Expression
1		ssdifss	⊢ ( 𝐴 ⊆ ℝ* → ( 𝐴 ∖ { -∞ } ) ⊆ ℝ* )
2		supxrmnf	⊢ ( ( 𝐴 ∖ { -∞ } ) ⊆ ℝ* → sup ( ( ( 𝐴 ∖ { -∞ } ) ∪ { -∞ } ) , ℝ* , < ) = sup ( ( 𝐴 ∖ { -∞ } ) , ℝ* , < ) )
3	1 2	syl	⊢ ( 𝐴 ⊆ ℝ* → sup ( ( ( 𝐴 ∖ { -∞ } ) ∪ { -∞ } ) , ℝ* , < ) = sup ( ( 𝐴 ∖ { -∞ } ) , ℝ* , < ) )
4	3	adantr	⊢ ( ( 𝐴 ⊆ ℝ* ∧ -∞ ∈ 𝐴 ) → sup ( ( ( 𝐴 ∖ { -∞ } ) ∪ { -∞ } ) , ℝ* , < ) = sup ( ( 𝐴 ∖ { -∞ } ) , ℝ* , < ) )
5		difsnid	⊢ ( -∞ ∈ 𝐴 → ( ( 𝐴 ∖ { -∞ } ) ∪ { -∞ } ) = 𝐴 )
6	5	supeq1d	⊢ ( -∞ ∈ 𝐴 → sup ( ( ( 𝐴 ∖ { -∞ } ) ∪ { -∞ } ) , ℝ* , < ) = sup ( 𝐴 , ℝ* , < ) )
7	6	adantl	⊢ ( ( 𝐴 ⊆ ℝ* ∧ -∞ ∈ 𝐴 ) → sup ( ( ( 𝐴 ∖ { -∞ } ) ∪ { -∞ } ) , ℝ* , < ) = sup ( 𝐴 , ℝ* , < ) )
8	4 7	eqtr3d	⊢ ( ( 𝐴 ⊆ ℝ* ∧ -∞ ∈ 𝐴 ) → sup ( ( 𝐴 ∖ { -∞ } ) , ℝ* , < ) = sup ( 𝐴 , ℝ* , < ) )
9		difsn	⊢ ( ¬ -∞ ∈ 𝐴 → ( 𝐴 ∖ { -∞ } ) = 𝐴 )
10	9	supeq1d	⊢ ( ¬ -∞ ∈ 𝐴 → sup ( ( 𝐴 ∖ { -∞ } ) , ℝ* , < ) = sup ( 𝐴 , ℝ* , < ) )
11	10	adantl	⊢ ( ( 𝐴 ⊆ ℝ* ∧ ¬ -∞ ∈ 𝐴 ) → sup ( ( 𝐴 ∖ { -∞ } ) , ℝ* , < ) = sup ( 𝐴 , ℝ* , < ) )
12	8 11	pm2.61dan	⊢ ( 𝐴 ⊆ ℝ* → sup ( ( 𝐴 ∖ { -∞ } ) , ℝ* , < ) = sup ( 𝐴 , ℝ* , < ) )