Theorem wfii

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)

	Ref	Expression
Hypotheses	wfi.1	|- R We A
	wfi.2	|- R Se A
Assertion	wfii	|- ( ( B C_ A /\ A. y e. A ( Pred ( R , A , y ) C_ B -> y e. 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 C_ A /\ A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) ) -> A = B )
4	1 2 3	mpanl12	|- ( ( B C_ A /\ A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) -> A = B )