Theorem weinxp

Description: Intersection of well-ordering with Cartesian product of its field. (Contributed by NM, 9-Mar-1997) (Revised by Mario Carneiro, 10-Jul-2014)

	Ref	Expression
Assertion	weinxp	⊢ ( 𝑅 We 𝐴 ↔ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) We 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		frinxp	⊢ ( 𝑅 Fr 𝐴 ↔ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) Fr 𝐴 )
2		soinxp	⊢ ( 𝑅 Or 𝐴 ↔ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) Or 𝐴 )
3	1 2	anbi12i	⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴 ) ↔ ( ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) Fr 𝐴 ∧ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) Or 𝐴 ) )
4		df-we	⊢ ( 𝑅 We 𝐴 ↔ ( 𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴 ) )
5		df-we	⊢ ( ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) We 𝐴 ↔ ( ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) Fr 𝐴 ∧ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) Or 𝐴 ) )
6	3 4 5	3bitr4i	⊢ ( 𝑅 We 𝐴 ↔ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) We 𝐴 )