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Theorem frc 5648
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 5642 . . . 4 (((𝐵 ∈ V ∧ 𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑧𝐵 ¬ 𝑧𝑅𝑥)
31, 2mpanl1 700 . . 3 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑧𝐵 ¬ 𝑧𝑅𝑥)
433impb 1115 . 2 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑧𝐵 ¬ 𝑧𝑅𝑥)
5 breq1 5146 . . . 4 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
65rabeq0w 4387 . . 3 ({𝑦𝐵𝑦𝑅𝑥} = ∅ ↔ ∀𝑧𝐵 ¬ 𝑧𝑅𝑥)
76rexbii 3094 . 2 (∃𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} = ∅ ↔ ∃𝑥𝐵𝑧𝐵 ¬ 𝑧𝑅𝑥)
84, 7sylibr 234 1 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  wss 3951  c0 4333   class class class wbr 5143   Fr wfr 5634
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-fr 5637
This theorem is referenced by:  frirr  5661  epfrc  5670
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