This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 → 𝑦 ∈ 𝐵 ) ) ) → 𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 wefr ⊢ ( 𝑅 We 𝐴 → 𝑅 Fr 𝐴 )
2 1 adantr ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Fr 𝐴 )
3 weso ⊢ ( 𝑅 We 𝐴 → 𝑅 Or 𝐴 )
4 sopo ⊢ ( 𝑅 Or 𝐴 → 𝑅 Po 𝐴 )
5 3 4 syl ⊢ ( 𝑅 We 𝐴 → 𝑅 Po 𝐴 )
6 5 adantr ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Po 𝐴 )
7 simpr ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Se 𝐴 )
8 2 6 7 3jca ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) )
9 frpoind ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 → 𝑦 ∈ 𝐵 ) ) ) → 𝐴 = 𝐵 )
10 8 9 sylan ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 → 𝑦 ∈ 𝐵 ) ) ) → 𝐴 = 𝐵 )