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	$$⊢R\mathrm{We}A$$
	wfi.2	$$⊢R\mathrm{Se}A$$
Assertion	wfii	$$⊢\left(B\subseteq A\wedge \forall y\in A\left(Pred\left(R,A,y\right)\subseteq B\to y\in B\right)\right)\to A=B$$

Proof

Step	Hyp	Ref	Expression
1		wfi.1	$$⊢R\mathrm{We}A$$
2		wfi.2	$$⊢R\mathrm{Se}A$$
3		wfi	$$⊢\left(\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\wedge \left(B\subseteq A\wedge \forall y\in A\left(Pred\left(R,A,y\right)\subseteq B\to y\in B\right)\right)\right)\to A=B$$
4	1 2 3	mpanl12	$$⊢\left(B\subseteq A\wedge \forall y\in A\left(Pred\left(R,A,y\right)\subseteq B\to y\in B\right)\right)\to A=B$$