Theorem isoso

Description: An isomorphism preserves the property of being a strict total order. (Contributed by Stefan O'Rear, 16-Nov-2014)

	Ref	Expression
Assertion	isoso	⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		isocnv	⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ◡ 𝐻 Isom 𝑆 , 𝑅 ( 𝐵 , 𝐴 ) )
2		isosolem	⊢ ( ◡ 𝐻 Isom 𝑆 , 𝑅 ( 𝐵 , 𝐴 ) → ( 𝑅 Or 𝐴 → 𝑆 Or 𝐵 ) )
3	1 2	syl	⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑅 Or 𝐴 → 𝑆 Or 𝐵 ) )
4		isosolem	⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑆 Or 𝐵 → 𝑅 Or 𝐴 ) )
5	3 4	impbid	⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵 ) )