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Theorem fr0 4329
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 4310 . 2 (𝑅 Fr ∅ ↔ ∀𝑠 FrFor 𝑅𝑠)
2 0ss 3447 . . . 4 ∅ ⊆ 𝑠
32a1i 9 . . 3 (∀𝑥 ∈ ∅ (∀𝑦 ∈ ∅ (𝑦𝑅𝑥𝑦𝑠) → 𝑥𝑠) → ∅ ⊆ 𝑠)
4 df-frfor 4309 . . 3 ( FrFor 𝑅𝑠 ↔ (∀𝑥 ∈ ∅ (∀𝑦 ∈ ∅ (𝑦𝑅𝑥𝑦𝑠) → 𝑥𝑠) → ∅ ⊆ 𝑠))
53, 4mpbir 145 . 2 FrFor 𝑅𝑠
61, 5mpgbir 1441 1 𝑅 Fr ∅
Colors of variables: wff set class
Syntax hints:  wi 4  wral 2444  wss 3116  c0 3409   class class class wbr 3982   FrFor wfrfor 4305   Fr wfr 4306
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-frfor 4309  df-frind 4310
This theorem is referenced by:  we0  4339
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