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Theorem frc 5508
Description: Property of well-founded relation (one direction of definition using class variables). (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 19-Nov-2014.)
Hypothesis
Ref Expression
frc.1 𝐵 ∈ V
Assertion
Ref Expression
frc ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} = ∅)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Proof of Theorem frc
StepHypRef Expression
1 frc.1 . . . 4 𝐵 ∈ V
2 fri 5504 . . . 4 (((𝐵 ∈ V ∧ 𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
31, 2mpanl1 699 . . 3 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
433impb 1112 . 2 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
5 rabeq0 4321 . . 3 ({𝑦𝐵𝑦𝑅𝑥} = ∅ ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
65rexbii 3241 . 2 (∃𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} = ∅ ↔ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
74, 6sylibr 237 1 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wral 3133  wrex 3134  {crab 3137  Vcvv 3480  wss 3919  c0 4276   class class class wbr 5052   Fr wfr 5498
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-in 3926  df-ss 3936  df-nul 4277  df-fr 5501
This theorem is referenced by:  frirr  5519  epfrc  5528
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