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Theorem frc 5597
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 5591 . . . 4 (((𝐵 ∈ V ∧ 𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑧𝐵 ¬ 𝑧𝑅𝑥)
31, 2mpanl1 698 . . 3 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑧𝐵 ¬ 𝑧𝑅𝑥)
433impb 1115 . 2 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑧𝐵 ¬ 𝑧𝑅𝑥)
5 breq1 5106 . . . 4 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
65rabeq0w 4341 . . 3 ({𝑦𝐵𝑦𝑅𝑥} = ∅ ↔ ∀𝑧𝐵 ¬ 𝑧𝑅𝑥)
76rexbii 3095 . 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 396  w3a 1087   = wceq 1541  wcel 2106  wne 2941  wral 3062  wrex 3071  {crab 3405  Vcvv 3443  wss 3908  c0 4280   class class class wbr 5103   Fr wfr 5583
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-fr 5586
This theorem is referenced by:  frirr  5608  epfrc  5617
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