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Theorem relsnb 5815
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 5814 . . . 4 (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
21biimpcd 249 . . 3 (Rel {𝐴} → (𝐴 ∈ V → 𝐴 ∈ (V × V)))
3 imor 853 . . 3 ((𝐴 ∈ V → 𝐴 ∈ (V × V)) ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
42, 3sylib 218 . 2 (Rel {𝐴} → (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
5 snprc 4722 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
6 rel0 5812 . . . . 5 Rel ∅
7 releq 5789 . . . . 5 ({𝐴} = ∅ → (Rel {𝐴} ↔ Rel ∅))
86, 7mpbiri 258 . . . 4 ({𝐴} = ∅ → Rel {𝐴})
95, 8sylbi 217 . . 3 𝐴 ∈ V → Rel {𝐴})
10 relsng 5814 . . . 4 (𝐴 ∈ (V × V) → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
1110ibir 268 . . 3 (𝐴 ∈ (V × V) → Rel {𝐴})
129, 11jaoi 857 . 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 847   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  {csn 4631   × cxp 5687  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-ss 3980  df-nul 4340  df-sn 4632  df-rel 5696
This theorem is referenced by: (None)
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