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Theorem fr0 5645
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 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffr2 5630 . 2 (𝑅 Fr ∅ ↔ ∀𝑥((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
2 ss0 4390 . . . . 5 (𝑥 ⊆ ∅ → 𝑥 = ∅)
32a1d 25 . . . 4 (𝑥 ⊆ ∅ → (¬ ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅ → 𝑥 = ∅))
43necon1ad 2949 . . 3 (𝑥 ⊆ ∅ → (𝑥 ≠ ∅ → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
54imp 406 . 2 ((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅)
61, 5mpgbir 1793 1 𝑅 Fr ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1533  wne 2932  wrex 3062  {crab 3424  wss 3940  c0 4314   class class class wbr 5138   Fr wfr 5618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-fr 5621
This theorem is referenced by:  we0  5661  frsn  5753  frfi  9284  ifr0  43698
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