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Theorem wefr 4287
Description: A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)
Ref Expression
wefr (𝑅 We 𝐴𝑅 Fr 𝐴)

Proof of Theorem wefr
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wetr 4263 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
21simplbi 272 1 (𝑅 We 𝐴𝑅 Fr 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wral 2417   class class class wbr 3936   Fr wfr 4257   We wwe 4259
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 4263
This theorem is referenced by:  wepo  4288  wetriext  4498
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