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Theorem fri 5211
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 5208 . . 3 (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥))
2 sseq1 3775 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝐴𝐵𝐴))
3 neeq1 3005 . . . . . 6 (𝑧 = 𝐵 → (𝑧 ≠ ∅ ↔ 𝐵 ≠ ∅))
42, 3anbi12d 616 . . . . 5 (𝑧 = 𝐵 → ((𝑧𝐴𝑧 ≠ ∅) ↔ (𝐵𝐴𝐵 ≠ ∅)))
5 raleq 3287 . . . . . 6 (𝑧 = 𝐵 → (∀𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
65rexeqbi1dv 3296 . . . . 5 (𝑧 = 𝐵 → (∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
74, 6imbi12d 333 . . . 4 (𝑧 = 𝐵 → (((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥) ↔ ((𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
87spcgv 3444 . . 3 (𝐵𝐶 → (∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥) → ((𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
91, 8syl5bi 232 . 2 (𝐵𝐶 → (𝑅 Fr 𝐴 → ((𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
109imp31 404 1 (((𝐵𝐶𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wal 1629   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  wss 3723  c0 4063   class class class wbr 4786   Fr wfr 5205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-fr 5208
This theorem is referenced by:  frc  5215  fr2nr  5227  frminex  5229  wereu  5245  wereu2  5246  fr3nr  7126  frfi  8361  fimax2g  8362  fimin2g  8559  wofib  8606  wemapso  8612  wemapso2lem  8613  noinfep  8721  cflim2  9287  isfin1-3  9410  fin12  9437  fpwwe2lem12  9665  fpwwe2lem13  9666  fpwwe2  9667  bnj110  31266  frpomin  32075  frinfm  33862  fdc  33873  fnwe2lem2  38147
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