Theorem wfii

Description: The Principle of Well-Founded Induction. Theorem 6.27 of TakeutiZaring p. 32. This principle states that if B is a subclass of a well-ordered class A with the property that every element of B whose inital segment is included in A is itself equal to A . (Contributed by Scott Fenton, 29-Jan-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	wfi.1	⊢ 𝑅 We 𝐴
	wfi.2	⊢ 𝑅 Se 𝐴
Assertion	wfii	⊢ ( ( 𝐵 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 → 𝑦 ∈ 𝐵 ) ) → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		wfi.1	⊢ 𝑅 We 𝐴
2		wfi.2	⊢ 𝑅 Se 𝐴
3		wfi	⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 → 𝑦 ∈ 𝐵 ) ) ) → 𝐴 = 𝐵 )
4	1 2 3	mpanl12	⊢ ( ( 𝐵 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 → 𝑦 ∈ 𝐵 ) ) → 𝐴 = 𝐵 )