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Theorem fr0 4243
Description: Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)
Assertion
Ref Expression
fr0 𝑅 Fr ∅

Proof of Theorem fr0
Dummy variables 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-frind 4224 . 2 (𝑅 Fr ∅ ↔ ∀𝑠 FrFor 𝑅𝑠)
2 0ss 3371 . . . 4 ∅ ⊆ 𝑠
32a1i 9 . . 3 (∀𝑥 ∈ ∅ (∀𝑦 ∈ ∅ (𝑦𝑅𝑥𝑦𝑠) → 𝑥𝑠) → ∅ ⊆ 𝑠)
4 df-frfor 4223 . . 3 ( FrFor 𝑅𝑠 ↔ (∀𝑥 ∈ ∅ (∀𝑦 ∈ ∅ (𝑦𝑅𝑥𝑦𝑠) → 𝑥𝑠) → ∅ ⊆ 𝑠))
53, 4mpbir 145 . 2 FrFor 𝑅𝑠
61, 5mpgbir 1414 1 𝑅 Fr ∅
Colors of variables: wff set class
Syntax hints:  wi 4  wral 2393  wss 3041  c0 3333   class class class wbr 3899   FrFor wfrfor 4219   Fr wfr 4220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-nul 3334  df-frfor 4223  df-frind 4224
This theorem is referenced by:  we0  4253
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