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Theorem cnvti 6906
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 4720 . . . . 5 ((𝑢𝐴𝑣𝐴) → (𝑢𝑅𝑣𝑣𝑅𝑢))
54notbid 656 . . . 4 ((𝑢𝐴𝑣𝐴) → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑣𝑅𝑢))
6 brcnvg 4720 . . . . . 6 ((𝑣𝐴𝑢𝐴) → (𝑣𝑅𝑢𝑢𝑅𝑣))
76ancoms 266 . . . . 5 ((𝑢𝐴𝑣𝐴) → (𝑣𝑅𝑢𝑢𝑅𝑣))
87notbid 656 . . . 4 ((𝑢𝐴𝑣𝐴) → (¬ 𝑣𝑅𝑢 ↔ ¬ 𝑢𝑅𝑣))
95, 8anbi12d 464 . . 3 ((𝑢𝐴𝑣𝐴) → ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣)))
109adantl 275 . 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 1480   class class class wbr 3929  ccnv 4538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547
This theorem is referenced by:  eqinfti  6907  infvalti  6909  infclti  6910  inflbti  6911  infglbti  6912  infmoti  6915  infsnti  6917  infisoti  6919  infrenegsupex  9401  infxrnegsupex  11044
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