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Theorem relsnb 5709
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 5708 . . . 4 (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
21biimpcd 248 . . 3 (Rel {𝐴} → (𝐴 ∈ V → 𝐴 ∈ (V × V)))
3 imor 849 . . 3 ((𝐴 ∈ V → 𝐴 ∈ (V × V)) ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
42, 3sylib 217 . 2 (Rel {𝐴} → (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
5 snprc 4658 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
6 rel0 5706 . . . . 5 Rel ∅
7 releq 5685 . . . . 5 ({𝐴} = ∅ → (Rel {𝐴} ↔ Rel ∅))
86, 7mpbiri 257 . . . 4 ({𝐴} = ∅ → Rel {𝐴})
95, 8sylbi 216 . . 3 𝐴 ∈ V → Rel {𝐴})
10 relsng 5708 . . . 4 (𝐴 ∈ (V × V) → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
1110ibir 267 . . 3 (𝐴 ∈ (V × V) → Rel {𝐴})
129, 11jaoi 853 . 2 ((¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)) → Rel {𝐴})
134, 12impbii 208 1 (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wo 843   = wceq 1541  wcel 2109  Vcvv 3430  c0 4261  {csn 4566   × cxp 5586  Rel wrel 5593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-v 3432  df-dif 3894  df-in 3898  df-ss 3908  df-nul 4262  df-sn 4567  df-rel 5595
This theorem is referenced by: (None)
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