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Theorem frc 5638
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
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 frc.1 . . . 4 𝐵 ∈ V
2 fri 5632 . . . 4 (((𝐵 ∈ V ∧ 𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑧𝐵 ¬ 𝑧𝑅𝑥)
31, 2mpanl1 699 . . 3 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑧𝐵 ¬ 𝑧𝑅𝑥)
433impb 1113 . 2 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑧𝐵 ¬ 𝑧𝑅𝑥)
5 breq1 5145 . . . 4 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
65rabeq0w 4379 . . 3 ({𝑦𝐵𝑦𝑅𝑥} = ∅ ↔ ∀𝑧𝐵 ¬ 𝑧𝑅𝑥)
76rexbii 3089 . 2 (∃𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} = ∅ ↔ ∃𝑥𝐵𝑧𝐵 ¬ 𝑧𝑅𝑥)
84, 7sylibr 233 1 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2935  wral 3056  wrex 3065  {crab 3427  Vcvv 3469  wss 3944  c0 4318   class class class wbr 5142   Fr wfr 5624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-fr 5627
This theorem is referenced by:  frirr  5649  epfrc  5658
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