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 C_ A /\ A. y e. A ( Pred ( R , A , y ) C_ B -> y e. 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 C_ A /\ A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) ) -> A = B )
10	8 9	sylan	|- ( ( ( R We A /\ R Se A ) /\ ( B C_ A /\ A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) ) -> A = B )