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Theorem dffr2 5639
Description: Alternate definition of well-founded relation. Similar to Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Mario Carneiro, 23-Jun-2015.) Avoid ax-10 2137, ax-11 2154, ax-12 2171, but use ax-8 2108. (Revised by Gino Giotto, 3-Oct-2024.)
Assertion
Ref Expression
dffr2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝑅,𝑦,𝑧

Proof of Theorem dffr2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-fr 5630 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑤𝑥 ¬ 𝑤𝑅𝑦))
2 breq1 5150 . . . . . 6 (𝑧 = 𝑤 → (𝑧𝑅𝑦𝑤𝑅𝑦))
32rabeq0w 4382 . . . . 5 ({𝑧𝑥𝑧𝑅𝑦} = ∅ ↔ ∀𝑤𝑥 ¬ 𝑤𝑅𝑦)
43rexbii 3094 . . . 4 (∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅ ↔ ∃𝑦𝑥𝑤𝑥 ¬ 𝑤𝑅𝑦)
54imbi2i 335 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑤𝑥 ¬ 𝑤𝑅𝑦))
65albii 1821 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑤𝑥 ¬ 𝑤𝑅𝑦))
71, 6bitr4i 277 1 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wne 2940  wral 3061  wrex 3070  {crab 3432  wss 3947  c0 4321   class class class wbr 5147   Fr wfr 5627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-fr 5630
This theorem is referenced by:  fr0  5654  dfepfr  5660  dffr3  6095
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