Theorem papcotr 7607

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 4132	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋𝑅𝑧 ↔ 𝑋𝑅𝑍))
3		breq2 4132	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑌𝑅𝑧 ↔ 𝑌𝑅𝑍))
4	2, 3	orbi12d 805	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑋𝑅𝑧 ∨ 𝑌𝑅𝑧) ↔ (𝑋𝑅𝑍 ∨ 𝑌𝑅𝑍)))
5	4	imbi2d 230	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑋𝑅𝑌 → (𝑋𝑅𝑧 ∨ 𝑌𝑅𝑧)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑌𝑅𝑍))))
6		breq2 4132	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋𝑅𝑦 ↔ 𝑋𝑅𝑌))
7		breq1 4131	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦𝑅𝑧 ↔ 𝑌𝑅𝑧))
8	7	orbi2d 802	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋𝑅𝑧 ∨ 𝑦𝑅𝑧) ↔ (𝑋𝑅𝑧 ∨ 𝑌𝑅𝑧)))
9	6, 8	imbi12d 234	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑦𝑅𝑧)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑧 ∨ 𝑌𝑅𝑧))))
10	9	ralbidv 2550	. . . 4 ⊢ (𝑦 = 𝑌 → (∀𝑧 ∈ 𝐴 (𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑦𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 (𝑋𝑅𝑌 → (𝑋𝑅𝑧 ∨ 𝑌𝑅𝑧))))
11		breq1 4131	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦))
12		breq1 4131	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑧 ↔ 𝑋𝑅𝑧))
13	12	orbi1d 803	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧) ↔ (𝑋𝑅𝑧 ∨ 𝑦𝑅𝑧)))
14	11, 13	imbi12d 234	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ↔ (𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑦𝑅𝑧))))
15	14	2ralbidv 2574	. . . . 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 7602	. . . . . . 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 538	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)))
20		papsym.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
21	15, 19, 20	rspcdva 2934	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑦𝑅𝑧)))
22		papsym.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
23	10, 21, 22	rspcdva 2934	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 (𝑋𝑅𝑌 → (𝑋𝑅𝑧 ∨ 𝑌𝑅𝑧)))
24		papcotr.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐴)
25	5, 23, 24	rspcdva 2934	. 2 ⊢ (𝜑 → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑌𝑅𝑍)))
26	1, 25	mpd 13	1 ⊢ (𝜑 → (𝑋𝑅𝑍 ∨ 𝑌𝑅𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220   class class class wbr 4128   × cxp 4770   Ap wap 7601

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 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-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-pap 7602

This theorem is referenced by:  aprlring  14583