Theorem papcotr 7561

Description: An apartness is cotransitive. (Contributed by Jim Kingdon, 28-May-2026.)

Hypotheses

Ref	Expression
papsym.r	⊢ (𝜑 → 𝑅 Ap 𝐴)
papsym.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
papsym.y	⊢ (𝜑 → 𝑌 ∈ 𝐴)
papsym.ap	⊢ (𝜑 → 𝑋𝑅𝑌)
papcotr.z	⊢ (𝜑 → 𝑍 ∈ 𝐴)

Assertion

Ref	Expression
papcotr	⊢ (𝜑 → (𝑋𝑅𝑍 ∨ 𝑌𝑅𝑍))

Proof of Theorem papcotr

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		papsym.ap	. 2 ⊢ (𝜑 → 𝑋𝑅𝑌)
2		breq2 4112	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋𝑅𝑧 ↔ 𝑋𝑅𝑍))
3		breq2 4112	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑌𝑅𝑧 ↔ 𝑌𝑅𝑍))
4	2, 3	orbi12d 801	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑋𝑅𝑧 ∨ 𝑌𝑅𝑧) ↔ (𝑋𝑅𝑍 ∨ 𝑌𝑅𝑍)))
5	4	imbi2d 230	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑋𝑅𝑌 → (𝑋𝑅𝑧 ∨ 𝑌𝑅𝑧)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑌𝑅𝑍))))
6		breq2 4112	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋𝑅𝑦 ↔ 𝑋𝑅𝑌))
7		breq1 4111	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦𝑅𝑧 ↔ 𝑌𝑅𝑧))
8	7	orbi2d 798	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋𝑅𝑧 ∨ 𝑦𝑅𝑧) ↔ (𝑋𝑅𝑧 ∨ 𝑌𝑅𝑧)))
9	6, 8	imbi12d 234	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑦𝑅𝑧)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑧 ∨ 𝑌𝑅𝑧))))
10	9	ralbidv 2542	. . . 4 ⊢ (𝑦 = 𝑌 → (∀𝑧 ∈ 𝐴 (𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑦𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 (𝑋𝑅𝑌 → (𝑋𝑅𝑧 ∨ 𝑌𝑅𝑧))))
11		breq1 4111	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦))
12		breq1 4111	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑧 ↔ 𝑋𝑅𝑧))
13	12	orbi1d 799	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧) ↔ (𝑋𝑅𝑧 ∨ 𝑦𝑅𝑧)))
14	11, 13	imbi12d 234	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ↔ (𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑦𝑅𝑧))))
15	14	2ralbidv 2566	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑦𝑅𝑧))))
16		papsym.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 Ap 𝐴)
17		df-pap 7558	. . . . . . 7 ⊢ (𝑅 Ap 𝐴 ↔ ((𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)))))
18	16, 17	sylib 122	. . . . . 6 ⊢ (𝜑 → ((𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)))))
19	18	simprrd 534	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)))
20		papsym.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
21	15, 19, 20	rspcdva 2925	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑦𝑅𝑧)))
22		papsym.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
23	10, 21, 22	rspcdva 2925	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 (𝑋𝑅𝑌 → (𝑋𝑅𝑧 ∨ 𝑌𝑅𝑧)))
24		papcotr.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐴)
25	5, 23, 24	rspcdva 2925	. 2 ⊢ (𝜑 → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑌𝑅𝑍)))
26	1, 25	mpd 13	1 ⊢ (𝜑 → (𝑋𝑅𝑍 ∨ 𝑌𝑅𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398   ∈ wcel 2203  ∀wral 2520   ⊆ wss 3210   class class class wbr 4108   × cxp 4746   Ap wap 7557

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2814  df-un 3214  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-pap 7558

This theorem is referenced by:  aprlring  14426