Metamath Proof Explorer


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 RWeARSeABAyAPredRAyByBA=B

Proof

Step Hyp Ref Expression
1 wefr RWeARFrA
2 1 adantr RWeARSeARFrA
3 weso RWeAROrA
4 sopo ROrARPoA
5 3 4 syl RWeARPoA
6 5 adantr RWeARSeARPoA
7 simpr RWeARSeARSeA
8 2 6 7 3jca RWeARSeARFrARPoARSeA
9 frpoind RFrARPoARSeABAyAPredRAyByBA=B
10 8 9 sylan RWeARSeABAyAPredRAyByBA=B