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Metamath Proof Explorer

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	$⊢ (B \subseteq A \land \forall y \in A (Pred (R, A, y) \subseteq B \to y \in B)) \to A = B$

Proof

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