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Theorem dffr2 5641
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 2135, ax-11 2152, ax-12 2169, but use ax-8 2106. (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 5632 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑤𝑥 ¬ 𝑤𝑅𝑦))
2 breq1 5152 . . . . . 6 (𝑧 = 𝑤 → (𝑧𝑅𝑦𝑤𝑅𝑦))
32rabeq0w 4384 . . . . 5 ({𝑧𝑥𝑧𝑅𝑦} = ∅ ↔ ∀𝑤𝑥 ¬ 𝑤𝑅𝑦)
43rexbii 3092 . . . 4 (∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅ ↔ ∃𝑦𝑥𝑤𝑥 ¬ 𝑤𝑅𝑦)
54imbi2i 335 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑤𝑥 ¬ 𝑤𝑅𝑦))
65albii 1819 . 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 394  wal 1537   = wceq 1539  wne 2938  wral 3059  wrex 3068  {crab 3430  wss 3949  c0 4323   class class class wbr 5149   Fr wfr 5629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-fr 5632
This theorem is referenced by:  fr0  5656  dfepfr  5662  dffr3  6099
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