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Theorem fr0 5245
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 5231 . 2 (𝑅 Fr ∅ ↔ ∀𝑥((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
2 ss0 4117 . . . . 5 (𝑥 ⊆ ∅ → 𝑥 = ∅)
32a1d 25 . . . 4 (𝑥 ⊆ ∅ → (¬ ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅ → 𝑥 = ∅))
43necon1ad 2949 . . 3 (𝑥 ⊆ ∅ → (𝑥 ≠ ∅ → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
54imp 444 . 2 ((𝑥 ⊆ ∅ ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅)
61, 5mpgbir 1875 1 𝑅 Fr ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wne 2932  wrex 3051  {crab 3054  wss 3715  c0 4058   class class class wbr 4804   Fr wfr 5222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-in 3722  df-ss 3729  df-nul 4059  df-fr 5225
This theorem is referenced by:  we0  5261  frsn  5346  frfi  8372  ifr0  39174
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