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Theorem fr0 5662
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 5645 . 2 (𝑅 Fr ∅ ↔ ∀𝑥((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
2 ss0 4401 . . . . 5 (𝑥 ⊆ ∅ → 𝑥 = ∅)
32a1d 25 . . . 4 (𝑥 ⊆ ∅ → (¬ ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅ → 𝑥 = ∅))
43necon1ad 2956 . . 3 (𝑥 ⊆ ∅ → (𝑥 ≠ ∅ → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
54imp 406 . 2 ((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅)
61, 5mpgbir 1798 1 𝑅 Fr ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1539  wne 2939  wrex 3069  {crab 3435  wss 3950  c0 4332   class class class wbr 5142   Fr wfr 5633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-fr 5636
This theorem is referenced by:  we0  5679  frsn  5772  frfi  9322  ifr0  44474
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