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

Proof

Step	Hyp	Ref	Expression
1		wefr	$$⊢R\mathrm{We}A\to R\mathrm{Fr}A$$
2	1	adantr	$$⊢\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\to R\mathrm{Fr}A$$
3		weso	$$⊢R\mathrm{We}A\to R\mathrm{Or}A$$
4		sopo	$$⊢R\mathrm{Or}A\to R\mathrm{Po}A$$
5	3 4	syl	$$⊢R\mathrm{We}A\to R\mathrm{Po}A$$
6	5	adantr	$$⊢\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\to R\mathrm{Po}A$$
7		simpr	$$⊢\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\to R\mathrm{Se}A$$
8	2 6 7	3jca	$$⊢\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\to \left(R\mathrm{Fr}A\wedge R\mathrm{Po}A\wedge R\mathrm{Se}A\right)$$
9		frpoind	$$⊢\left(\left(R\mathrm{Fr}A\wedge R\mathrm{Po}A\wedge R\mathrm{Se}A\right)\wedge \left(B\subseteq A\wedge \forall y\in A\left(Pred\left(R,A,y\right)\subseteq B\to y\in B\right)\right)\right)\to A=B$$
10	8 9	sylan	$$⊢\left(\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\wedge \left(B\subseteq A\wedge \forall y\in A\left(Pred\left(R,A,y\right)\subseteq B\to y\in B\right)\right)\right)\to A=B$$