Theorem isowe

Description: An isomorphism preserves the property of being a well-ordering. Proposition 6.32(3) of TakeutiZaring p. 33. (Contributed by NM, 30-Apr-2004) (Revised by Mario Carneiro, 18-Nov-2014)

	Ref	Expression
Assertion	isowe	⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑅 We 𝐴 ↔ 𝑆 We 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		isofr	⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵 ) )
2		isoso	⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵 ) )
3	1 2	anbi12d	⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴 ) ↔ ( 𝑆 Fr 𝐵 ∧ 𝑆 Or 𝐵 ) ) )
4		df-we	⊢ ( 𝑅 We 𝐴 ↔ ( 𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴 ) )
5		df-we	⊢ ( 𝑆 We 𝐵 ↔ ( 𝑆 Fr 𝐵 ∧ 𝑆 Or 𝐵 ) )
6	3 4 5	3bitr4g	⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑅 We 𝐴 ↔ 𝑆 We 𝐵 ) )