Theorem relsnb 5639

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 5638	. . . 4 ⊢ (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
2	1	biimpcd 252	. . 3 ⊢ (Rel {𝐴} → (𝐴 ∈ V → 𝐴 ∈ (V × V)))
3		imor 850	. . 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 4613	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
6		rel0 5636	. . . . 5 ⊢ Rel ∅
7		releq 5615	. . . . 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 5638	. . . 4 ⊢ (𝐴 ∈ (V × V) → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
11	10	ibir 271	. . 3 ⊢ (𝐴 ∈ (V × V) → Rel {𝐴})
12	9, 11	jaoi 854	. 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 844   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ∅c0 4243  {csn 4525   × cxp 5517  Rel wrel 5524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-rel 5526

This theorem is referenced by: (None)