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Theorem fr0 4114
 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 4095 . 2 (𝑅 Fr ∅ ↔ ∀𝑠 FrFor 𝑅𝑠)
2 0ss 3289 . . . 4 ∅ ⊆ 𝑠
32a1i 9 . . 3 (∀𝑥 ∈ ∅ (∀𝑦 ∈ ∅ (𝑦𝑅𝑥𝑦𝑠) → 𝑥𝑠) → ∅ ⊆ 𝑠)
4 df-frfor 4094 . . 3 ( FrFor 𝑅𝑠 ↔ (∀𝑥 ∈ ∅ (∀𝑦 ∈ ∅ (𝑦𝑅𝑥𝑦𝑠) → 𝑥𝑠) → ∅ ⊆ 𝑠))
53, 4mpbir 144 . 2 FrFor 𝑅𝑠
61, 5mpgbir 1383 1 𝑅 Fr ∅
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wral 2349   ⊆ wss 2974  ∅c0 3258   class class class wbr 3793   FrFor wfrfor 4090   Fr wfr 4091 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-in 2980  df-ss 2987  df-nul 3259  df-frfor 4094  df-frind 4095 This theorem is referenced by:  we0  4124
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