Theorem relsnb 5806

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 5805	. . . 4 ⊢ (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
2	1	biimpcd 248	. . 3 ⊢ (Rel {𝐴} → (𝐴 ∈ V → 𝐴 ∈ (V × V)))
3		imor 851	. . 3 ⊢ ((𝐴 ∈ V → 𝐴 ∈ (V × V)) ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
4	2, 3	sylib 217	. 2 ⊢ (Rel {𝐴} → (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
5		snprc 4724	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
6		rel0 5803	. . . . 5 ⊢ Rel ∅
7		releq 5780	. . . . 5 ⊢ ({𝐴} = ∅ → (Rel {𝐴} ↔ Rel ∅))
8	6, 7	mpbiri 257	. . . 4 ⊢ ({𝐴} = ∅ → Rel {𝐴})
9	5, 8	sylbi 216	. . 3 ⊢ (¬ 𝐴 ∈ V → Rel {𝐴})
10		relsng 5805	. . . 4 ⊢ (𝐴 ∈ (V × V) → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
11	10	ibir 267	. . 3 ⊢ (𝐴 ∈ (V × V) → Rel {𝐴})
12	9, 11	jaoi 855	. 2 ⊢ ((¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)) → Rel {𝐴})
13	4, 12	impbii 208	1 ⊢ (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 845   = wceq 1533   ∈ wcel 2098  Vcvv 3471  ∅c0 4324  {csn 4630   × cxp 5678  Rel wrel 5685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3473  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4325  df-sn 4631  df-rel 5687

This theorem is referenced by: (None)