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 867	. . 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 4680	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
6		rel0 5785	. . . . 5 ⊢ Rel ∅
7		releq 5761	. . . . 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 871	. 2 ⊢ ((¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)) → Rel {𝐴})
13	4, 12	impbii 212	1 ⊢ (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ∅c0 4281  {csn 4588   × cxp 5656  Rel wrel 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-dif 4050  df-nul 4282  df-sn 4589  df-rel 5665

This theorem is used by: (None)