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Theorem dffr2 5645
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 2140, ax-11 2156, ax-12 2176, but use ax-8 2109. (Revised by GG, 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 5636 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑤𝑥 ¬ 𝑤𝑅𝑦))
2 breq1 5145 . . . . . 6 (𝑧 = 𝑤 → (𝑧𝑅𝑦𝑤𝑅𝑦))
32rabeq0w 4386 . . . . 5 ({𝑧𝑥𝑧𝑅𝑦} = ∅ ↔ ∀𝑤𝑥 ¬ 𝑤𝑅𝑦)
43rexbii 3093 . . . 4 (∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅ ↔ ∃𝑦𝑥𝑤𝑥 ¬ 𝑤𝑅𝑦)
54imbi2i 336 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑤𝑥 ¬ 𝑤𝑅𝑦))
65albii 1818 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑤𝑥 ¬ 𝑤𝑅𝑦))
71, 6bitr4i 278 1 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1537   = wceq 1539  wne 2939  wral 3060  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-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:  fr0  5662  dfepfr  5668  dffr3  6116
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