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Theorem dffr2 5044
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.)
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
StepHypRef Expression
1 df-fr 5038 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
2 rabeq0 3936 . . . . 5 ({𝑧𝑥𝑧𝑅𝑦} = ∅ ↔ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦)
32rexbii 3035 . . . 4 (∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅ ↔ ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)
43imbi2i 326 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
54albii 1744 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
61, 5bitr4i 267 1 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wne 2790  wral 2907  wrex 2908  {crab 2911  wss 3559  c0 3896   class class class wbr 4618   Fr wfr 5035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-nul 3897  df-fr 5038
This theorem is referenced by:  fr0  5058  dfepfr  5064  dffr3  5462
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