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Theorem cnvti 6821
Description: If a relation satisfies a condition corresponding to tightness of an apartness generated by an order, so does its converse. (Contributed by Jim Kingdon, 17-Dec-2021.)
Hypothesis
Ref Expression
eqinfti.ti ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
Assertion
Ref Expression
cnvti ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
Distinct variable groups:   𝑢,𝐴,𝑣   𝜑,𝑢,𝑣   𝑢,𝑅,𝑣

Proof of Theorem cnvti
StepHypRef Expression
1 eqinfti.ti . . 3 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
2 ancom 264 . . 3 ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣))
31, 2syl6bb 195 . 2 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣)))
4 brcnvg 4658 . . . . 5 ((𝑢𝐴𝑣𝐴) → (𝑢𝑅𝑣𝑣𝑅𝑢))
54notbid 633 . . . 4 ((𝑢𝐴𝑣𝐴) → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑣𝑅𝑢))
6 brcnvg 4658 . . . . . 6 ((𝑣𝐴𝑢𝐴) → (𝑣𝑅𝑢𝑢𝑅𝑣))
76ancoms 266 . . . . 5 ((𝑢𝐴𝑣𝐴) → (𝑣𝑅𝑢𝑢𝑅𝑣))
87notbid 633 . . . 4 ((𝑢𝐴𝑣𝐴) → (¬ 𝑣𝑅𝑢 ↔ ¬ 𝑢𝑅𝑣))
95, 8anbi12d 460 . . 3 ((𝑢𝐴𝑣𝐴) → ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣)))
109adantl 273 . 2 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣)))
113, 10bitr4d 190 1 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wcel 1448   class class class wbr 3875  ccnv 4476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-cnv 4485
This theorem is referenced by:  eqinfti  6822  infvalti  6824  infclti  6825  inflbti  6826  infglbti  6827  infmoti  6830  infsnti  6832  infisoti  6834  infrenegsupex  9239  infxrnegsupex  10871
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