Theorem wefr 4218

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 4194	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
2	1	simplbi 270	1 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)

Syntax hints:   → wi 4   ∧ wa 103  ∀wral 2375   class class class wbr 3875   Fr wfr 4188   We wwe 4190

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105

This theorem depends on definitions:  df-bi 116  df-wetr 4194

This theorem is referenced by:  wepo  4219  wetriext  4429