Theorem cnvti 6996

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 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))

Assertion

Ref	Expression
cnvti	⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢◡𝑅𝑣 ∧ ¬ 𝑣◡𝑅𝑢)))

Distinct variable groups:   𝑢,𝐴,𝑣   𝜑,𝑢,𝑣   𝑢,𝑅,𝑣

Proof of Theorem cnvti

Step	Hyp	Ref	Expression
1		eqinfti.ti	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
2		ancom 264	. . 3 ⊢ ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣))
3	1, 2	bitrdi 195	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣)))
4		brcnvg 4792	. . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢◡𝑅𝑣 ↔ 𝑣𝑅𝑢))
5	4	notbid 662	. . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (¬ 𝑢◡𝑅𝑣 ↔ ¬ 𝑣𝑅𝑢))
6		brcnvg 4792	. . . . . 6 ⊢ ((𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) → (𝑣◡𝑅𝑢 ↔ 𝑢𝑅𝑣))
7	6	ancoms 266	. . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑣◡𝑅𝑢 ↔ 𝑢𝑅𝑣))
8	7	notbid 662	. . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (¬ 𝑣◡𝑅𝑢 ↔ ¬ 𝑢𝑅𝑣))
9	5, 8	anbi12d 470	. . 3 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((¬ 𝑢◡𝑅𝑣 ∧ ¬ 𝑣◡𝑅𝑢) ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣)))
10	9	adantl 275	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((¬ 𝑢◡𝑅𝑣 ∧ ¬ 𝑣◡𝑅𝑢) ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣)))
11	3, 10	bitr4d 190	1 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢◡𝑅𝑣 ∧ ¬ 𝑣◡𝑅𝑢)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 2141   class class class wbr 3989  ^◡ccnv 4610

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619

This theorem is referenced by:  eqinfti  6997  infvalti  6999  infclti  7000  inflbti  7001  infglbti  7002  infmoti  7005  infsnti  7007  infisoti  7009  infrenegsupex  9553  infxrnegsupex  11226