Theorem relsnb 5759

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 5758	. . . 4 ⊢ (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
2	1	biimpcd 249	. . 3 ⊢ (Rel {𝐴} → (𝐴 ∈ V → 𝐴 ∈ (V × V)))
3		imor 854	. . 3 ⊢ ((𝐴 ∈ V → 𝐴 ∈ (V × V)) ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
4	2, 3	sylib 218	. 2 ⊢ (Rel {𝐴} → (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
5		snprc 4676	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
6		rel0 5756	. . . . 5 ⊢ Rel ∅
7		releq 5734	. . . . 5 ⊢ ({𝐴} = ∅ → (Rel {𝐴} ↔ Rel ∅))
8	6, 7	mpbiri 258	. . . 4 ⊢ ({𝐴} = ∅ → Rel {𝐴})
9	5, 8	sylbi 217	. . 3 ⊢ (¬ 𝐴 ∈ V → Rel {𝐴})
10		relsng 5758	. . . 4 ⊢ (𝐴 ∈ (V × V) → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
11	10	ibir 268	. . 3 ⊢ (𝐴 ∈ (V × V) → Rel {𝐴})
12	9, 11	jaoi 858	. 2 ⊢ ((¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)) → Rel {𝐴})
13	4, 12	impbii 209	1 ⊢ (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 848   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  {csn 4582   × cxp 5630  Rel wrel 5637

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-ss 3920  df-nul 4288  df-sn 4583  df-rel 5639

This theorem is referenced by: (None)