Theorem wess

Description: Subset theorem for the well-ordering predicate. Exercise 4 of TakeutiZaring p. 31. (Contributed by NM, 19-Apr-1994)

	Ref	Expression
Assertion	wess	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑅 We 𝐵 → 𝑅 We 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		frss	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑅 Fr 𝐵 → 𝑅 Fr 𝐴 ) )
2		soss	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑅 Or 𝐵 → 𝑅 Or 𝐴 ) )
3	1 2	anim12d	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵 ) → ( 𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴 ) ) )
4		df-we	⊢ ( 𝑅 We 𝐵 ↔ ( 𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵 ) )
5		df-we	⊢ ( 𝑅 We 𝐴 ↔ ( 𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴 ) )
6	3 4 5	3imtr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑅 We 𝐵 → 𝑅 We 𝐴 ) )