Theorem papirr 7559

Description: An apartness is irreflexive. (Contributed by Jim Kingdon, 27-May-2026.)

Assertion

Ref	Expression
papirr	⊢ ((𝑅 Ap 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋𝑅𝑋)

Proof of Theorem papirr

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

Step	Hyp	Ref	Expression
1		id 19	. . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
2	1, 1	breq12d 4121	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑥 ↔ 𝑋𝑅𝑋))
3	2	notbid 673	. 2 ⊢ (𝑥 = 𝑋 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑋𝑅𝑋))
4		df-pap 7558	. . . . 5 ⊢ (𝑅 Ap 𝐴 ↔ ((𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)))))
5	4	simplbi 274	. . . 4 ⊢ (𝑅 Ap 𝐴 → (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
6	5	simprd 114	. . 3 ⊢ (𝑅 Ap 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
7	6	adantr 276	. 2 ⊢ ((𝑅 Ap 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
8		simpr 110	. 2 ⊢ ((𝑅 Ap 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
9	3, 7, 8	rspcdva 2925	1 ⊢ ((𝑅 Ap 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋𝑅𝑋)

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-in1 619  ax-in2 620  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:  aprnzr  14425