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Theorem cnvti 6661
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 262 . . 3 ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣))
31, 2syl6bb 194 . 2 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣)))
4 brcnvg 4587 . . . . 5 ((𝑢𝐴𝑣𝐴) → (𝑢𝑅𝑣𝑣𝑅𝑢))
54notbid 625 . . . 4 ((𝑢𝐴𝑣𝐴) → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑣𝑅𝑢))
6 brcnvg 4587 . . . . . 6 ((𝑣𝐴𝑢𝐴) → (𝑣𝑅𝑢𝑢𝑅𝑣))
76ancoms 264 . . . . 5 ((𝑢𝐴𝑣𝐴) → (𝑣𝑅𝑢𝑢𝑅𝑣))
87notbid 625 . . . 4 ((𝑢𝐴𝑣𝐴) → (¬ 𝑣𝑅𝑢 ↔ ¬ 𝑢𝑅𝑣))
95, 8anbi12d 457 . . 3 ((𝑢𝐴𝑣𝐴) → ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣)))
109adantl 271 . 2 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣)))
113, 10bitr4d 189 1 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wcel 1436   class class class wbr 3822  ccnv 4412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-cnv 4421
This theorem is referenced by:  eqinfti  6662  infvalti  6664  infclti  6665  inflbti  6666  infglbti  6667  infmoti  6670  infsnti  6672  infisoti  6674  infrenegsupex  9017
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