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Theorem frc 5600
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 5594 . . . 4 (((𝐵 ∈ V ∧ 𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑧𝐵 ¬ 𝑧𝑅𝑥)
31, 2mpanl1 699 . . 3 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑧𝐵 ¬ 𝑧𝑅𝑥)
433impb 1116 . 2 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑧𝐵 ¬ 𝑧𝑅𝑥)
5 breq1 5109 . . . 4 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
65rabeq0w 4344 . . 3 ({𝑦𝐵𝑦𝑅𝑥} = ∅ ↔ ∀𝑧𝐵 ¬ 𝑧𝑅𝑥)
76rexbii 3098 . 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 397  w3a 1088   = wceq 1542  wcel 2107  wne 2944  wral 3065  wrex 3074  {crab 3408  Vcvv 3446  wss 3911  c0 4283   class class class wbr 5106   Fr wfr 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-fr 5589
This theorem is referenced by:  frirr  5611  epfrc  5620
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