Theorem papsym 7602

Description: An apartness is symmetric. (Contributed by Jim Kingdon, 27-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)

Assertion

Ref	Expression
papsym	⊢ (𝜑 → 𝑌𝑅𝑋)

Proof of Theorem papsym

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 4129	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝑅𝑦 ↔ 𝑋𝑅𝑌))
3		breq1 4128	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦𝑅𝑋 ↔ 𝑌𝑅𝑋))
4	2, 3	imbi12d 234	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝑅𝑦 → 𝑦𝑅𝑋) ↔ (𝑋𝑅𝑌 → 𝑌𝑅𝑋)))
5		breq1 4128	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦))
6		breq2 4129	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑋))
7	5, 6	imbi12d 234	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ (𝑋𝑅𝑦 → 𝑦𝑅𝑋)))
8	7	ralbidv 2550	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑋𝑅𝑦 → 𝑦𝑅𝑋)))
9		papsym.r	. . . . . 6 ⊢ (𝜑 → 𝑅 Ap 𝐴)
10		df-pap 7598	. . . . . 6 ⊢ (𝑅 Ap 𝐴 ↔ ((𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)))))
11	9, 10	sylib 122	. . . . 5 ⊢ (𝜑 → ((𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)))))
12	11	simprld 536	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥))
13		papsym.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
14	8, 12, 13	rspcdva 2934	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (𝑋𝑅𝑦 → 𝑦𝑅𝑋))
15		papsym.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
16	4, 14, 15	rspcdva 2934	. 2 ⊢ (𝜑 → (𝑋𝑅𝑌 → 𝑌𝑅𝑋))
17	1, 16	mpd 13	1 ⊢ (𝜑 → 𝑌𝑅𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220   class class class wbr 4125   × cxp 4767   Ap wap 7597

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 3711  df-pr 3712  df-op 3714  df-br 4126  df-pap 7598

This theorem is referenced by:  aprlring  14573