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Theorem fri 5036
Description: Property of well-founded relation (one direction of definition). (Contributed by NM, 18-Mar-1997.)
Assertion
Ref Expression
fri (((𝐵𝐶𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem fri
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-fr 5033 . . 3 (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥))
2 sseq1 3605 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝐴𝐵𝐴))
3 neeq1 2852 . . . . . 6 (𝑧 = 𝐵 → (𝑧 ≠ ∅ ↔ 𝐵 ≠ ∅))
42, 3anbi12d 746 . . . . 5 (𝑧 = 𝐵 → ((𝑧𝐴𝑧 ≠ ∅) ↔ (𝐵𝐴𝐵 ≠ ∅)))
5 raleq 3127 . . . . . 6 (𝑧 = 𝐵 → (∀𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
65rexeqbi1dv 3136 . . . . 5 (𝑧 = 𝐵 → (∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
74, 6imbi12d 334 . . . 4 (𝑧 = 𝐵 → (((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥) ↔ ((𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
87spcgv 3279 . . 3 (𝐵𝐶 → (∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥) → ((𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
91, 8syl5bi 232 . 2 (𝐵𝐶 → (𝑅 Fr 𝐴 → ((𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
109imp31 448 1 (((𝐵𝐶𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1478   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  wss 3555  c0 3891   class class class wbr 4613   Fr wfr 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-v 3188  df-in 3562  df-ss 3569  df-fr 5033
This theorem is referenced by:  frc  5040  fr2nr  5052  frminex  5054  wereu  5070  wereu2  5071  fr3nr  6926  frfi  8149  fimax2g  8150  fimin2g  8347  wofib  8394  wemapso  8400  wemapso2lem  8401  noinfep  8501  cflim2  9029  isfin1-3  9152  fin12  9179  fpwwe2lem12  9407  fpwwe2lem13  9408  fpwwe2  9409  bnj110  30633  frinfm  33159  fdc  33170  fnwe2lem2  37098
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