Theorem cnvti 7359

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 4961	. . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢◡𝑅𝑣 ↔ 𝑣𝑅𝑢))
5	4	notbid 677	. . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (¬ 𝑢◡𝑅𝑣 ↔ ¬ 𝑣𝑅𝑢))
6		brcnvg 4961	. . . . . 6 ⊢ ((𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) → (𝑣◡𝑅𝑢 ↔ 𝑢𝑅𝑣))
7	6	ancoms 268	. . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑣◡𝑅𝑢 ↔ 𝑢𝑅𝑣))
8	7	notbid 677	. . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (¬ 𝑣◡𝑅𝑢 ↔ ¬ 𝑢𝑅𝑣))
9	5, 8	anbi12d 477	. . 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)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2209   class class class wbr 4130  ^◡ccnv 4773

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-cnv 4782

This theorem is used by:  eqinfti  7360  infvalti  7362  infclti  7363  inflbti  7364  infglbti  7365  infmoti  7368  infsnti  7370  infisoti  7372  infrenegsupex  9994  infxrnegsupex  12029