Theorem cnvti 7094

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 266	. . 3 ⊢ ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣))
3	1, 2	bitrdi 196	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣)))
4		brcnvg 4848	. . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢◡𝑅𝑣 ↔ 𝑣𝑅𝑢))
5	4	notbid 668	. . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (¬ 𝑢◡𝑅𝑣 ↔ ¬ 𝑣𝑅𝑢))
6		brcnvg 4848	. . . . . 6 ⊢ ((𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) → (𝑣◡𝑅𝑢 ↔ 𝑢𝑅𝑣))
7	6	ancoms 268	. . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑣◡𝑅𝑢 ↔ 𝑢𝑅𝑣))
8	7	notbid 668	. . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (¬ 𝑣◡𝑅𝑢 ↔ ¬ 𝑢𝑅𝑣))
9	5, 8	anbi12d 473	. . 3 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((¬ 𝑢◡𝑅𝑣 ∧ ¬ 𝑣◡𝑅𝑢) ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣)))
10	9	adantl 277	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((¬ 𝑢◡𝑅𝑣 ∧ ¬ 𝑣◡𝑅𝑢) ↔ (¬ 𝑣𝑅𝑢 ∧ ¬ 𝑢𝑅𝑣)))
11	3, 10	bitr4d 191	1 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢◡𝑅𝑣 ∧ ¬ 𝑣◡𝑅𝑢)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2167   class class class wbr 4034  ^◡ccnv 4663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-cnv 4672

This theorem is referenced by:  eqinfti  7095  infvalti  7097  infclti  7098  inflbti  7099  infglbti  7100  infmoti  7103  infsnti  7105  infisoti  7107  infrenegsupex  9685  infxrnegsupex  11445