Theorem frindi

Description: The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin ). This principle states that if B is a subclass of a founded class A with the property that every element of B whose initial segment is included in A is itself equal to A . (Contributed by Scott Fenton, 6-Feb-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	frind.1	|- R Fr A
	frind.2	|- R Se A
Assertion	frindi	|- ( ( B C_ A /\ A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) -> A = B )

Proof

Step	Hyp	Ref	Expression
1		frind.1	|- R Fr A
2		frind.2	|- R Se A
3		frind	|- ( ( ( R Fr 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 )