Theorem wfi

Description: The Principle of Well-Ordered 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) (Proof shortened by Scott Fenton, 17-Nov-2024)

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

Proof

Step	Hyp	Ref	Expression
1		wefr	⊢ ( 𝑅 We 𝐴 → 𝑅 Fr 𝐴 )
2	1	adantr	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Fr 𝐴 )
3		weso	⊢ ( 𝑅 We 𝐴 → 𝑅 Or 𝐴 )
4		sopo	⊢ ( 𝑅 Or 𝐴 → 𝑅 Po 𝐴 )
5	3 4	syl	⊢ ( 𝑅 We 𝐴 → 𝑅 Po 𝐴 )
6	5	adantr	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Po 𝐴 )
7		simpr	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Se 𝐴 )
8	2 6 7	3jca	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) )
9		frpoind	⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 → 𝑦 ∈ 𝐵 ) ) ) → 𝐴 = 𝐵 )
10	8 9	sylan	⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 → 𝑦 ∈ 𝐵 ) ) ) → 𝐴 = 𝐵 )