Theorem wefr 4280

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 4256	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
2	1	simplbi 272	1 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)

Syntax hints:   → wi 4   ∧ wa 103  ∀wral 2416   class class class wbr 3929   Fr wfr 4250   We wwe 4252

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

This theorem depends on definitions:  df-bi 116  df-wetr 4256

This theorem is referenced by:  wepo  4281  wetriext  4491