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Theorem relsnb 5812
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
StepHypRef Expression
1 relsng 5811 . . . 4 (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
21biimpcd 249 . . 3 (Rel {𝐴} → (𝐴 ∈ V → 𝐴 ∈ (V × V)))
3 imor 854 . . 3 ((𝐴 ∈ V → 𝐴 ∈ (V × V)) ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
42, 3sylib 218 . 2 (Rel {𝐴} → (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
5 snprc 4717 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
6 rel0 5809 . . . . 5 Rel ∅
7 releq 5786 . . . . 5 ({𝐴} = ∅ → (Rel {𝐴} ↔ Rel ∅))
86, 7mpbiri 258 . . . 4 ({𝐴} = ∅ → Rel {𝐴})
95, 8sylbi 217 . . 3 𝐴 ∈ V → Rel {𝐴})
10 relsng 5811 . . . 4 (𝐴 ∈ (V × V) → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
1110ibir 268 . . 3 (𝐴 ∈ (V × V) → Rel {𝐴})
129, 11jaoi 858 . 2 ((¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)) → Rel {𝐴})
134, 12impbii 209 1 (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wo 848   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  {csn 4626   × cxp 5683  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-ss 3968  df-nul 4334  df-sn 4627  df-rel 5692
This theorem is referenced by: (None)
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