Theorem wefr 4505

Description: A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)

Assertion

Ref	Expression
wefr	⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)

Proof of Theorem wefr

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wetr 4481	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
2	1	simplbi 274	1 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  ∀wral 2528   class class class wbr 4131   Fr wfr 4475   We wwe 4477

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This proof depends on definitions:  df-bi 117  df-wetr 4481

This theorem is used by:  wepo  4506  wetriext  4726