Theorem relsnb 5789

Description: An at-most-singleton is a relation iff it is empty (because it is a "singleton on a proper class") or it is a singleton of an ordered pair. (Contributed by BJ, 26-Feb-2023.)

Assertion

Ref	Expression
relsnb	⊢ (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))

Proof of Theorem relsnb

Step	Hyp	Ref	Expression
1		relsng 5788	. . . 4 ⊢ (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
2	1	biimpcd 252	. . 3 ⊢ (Rel {𝐴} → (𝐴 ∈ V → 𝐴 ∈ (V × V)))
3		imor 866	. . 3 ⊢ ((𝐴 ∈ V → 𝐴 ∈ (V × V)) ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
4	2, 3	sylib 221	. 2 ⊢ (Rel {𝐴} → (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
5		snprc 4683	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
6		rel0 5785	. . . . 5 ⊢ Rel ∅
7		releq 5763	. . . . 5 ⊢ ({𝐴} = ∅ → (Rel {𝐴} ↔ Rel ∅))
8	6, 7	mpbiri 261	. . . 4 ⊢ ({𝐴} = ∅ → Rel {𝐴})
9	5, 8	sylbi 220	. . 3 ⊢ (¬ 𝐴 ∈ V → Rel {𝐴})
10		relsng 5788	. . . 4 ⊢ (𝐴 ∈ (V × V) → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
11	10	ibir 271	. . 3 ⊢ (𝐴 ∈ (V × V) → Rel {𝐴})
12	9, 11	jaoi 870	. 2 ⊢ ((¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)) → Rel {𝐴})
13	4, 12	impbii 212	1 ⊢ (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 860   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∅c0 4286  {csn 4589   × cxp 5659  Rel wrel 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-dif 3908  df-ss 3922  df-nul 4287  df-sn 4590  df-rel 5668

This theorem is referenced by: (None)