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Mirrors > Home > ILE Home > Th. List > wefr | GIF version |
Description: A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.) |
Ref | Expression |
---|---|
wefr | ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-wetr 4071 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | |
2 | 1 | simplbi 259 | 1 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wral 2306 class class class wbr 3764 Fr wfr 4065 We wwe 4067 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 |
This theorem depends on definitions: df-bi 110 df-wetr 4071 |
This theorem is referenced by: wepo 4096 wetriext 4301 |
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