Theorem negs11

Description: Surreal negation is one-to-one. (Contributed by Scott Fenton, 3-Feb-2025)

	Ref	Expression
Assertion	negs11	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( -us ‘ 𝐴 ) = ( -us ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( ( -us ‘ 𝐴 ) = ( -us ‘ 𝐵 ) → ( -us ‘ ( -us ‘ 𝐴 ) ) = ( -us ‘ ( -us ‘ 𝐵 ) ) )
2		negnegs	⊢ ( 𝐴 ∈ No → ( -us ‘ ( -us ‘ 𝐴 ) ) = 𝐴 )
3		negnegs	⊢ ( 𝐵 ∈ No → ( -us ‘ ( -us ‘ 𝐵 ) ) = 𝐵 )
4	2 3	eqeqan12d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( -us ‘ ( -us ‘ 𝐴 ) ) = ( -us ‘ ( -us ‘ 𝐵 ) ) ↔ 𝐴 = 𝐵 ) )
5	1 4	imbitrid	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( -us ‘ 𝐴 ) = ( -us ‘ 𝐵 ) → 𝐴 = 𝐵 ) )
6		fveq2	⊢ ( 𝐴 = 𝐵 → ( -us ‘ 𝐴 ) = ( -us ‘ 𝐵 ) )
7	5 6	impbid1	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( -us ‘ 𝐴 ) = ( -us ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )