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Theorem relsnb 5668
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 5667 . . . 4 (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
21biimpcd 251 . . 3 (Rel {𝐴} → (𝐴 ∈ V → 𝐴 ∈ (V × V)))
3 imor 849 . . 3 ((𝐴 ∈ V → 𝐴 ∈ (V × V)) ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
42, 3sylib 220 . 2 (Rel {𝐴} → (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
5 snprc 4645 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
6 rel0 5665 . . . . 5 Rel ∅
7 releq 5644 . . . . 5 ({𝐴} = ∅ → (Rel {𝐴} ↔ Rel ∅))
86, 7mpbiri 260 . . . 4 ({𝐴} = ∅ → Rel {𝐴})
95, 8sylbi 219 . . 3 𝐴 ∈ V → Rel {𝐴})
10 relsng 5667 . . . 4 (𝐴 ∈ (V × V) → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
1110ibir 270 . . 3 (𝐴 ∈ (V × V) → Rel {𝐴})
129, 11jaoi 853 . 2 ((¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)) → Rel {𝐴})
134, 12impbii 211 1 (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wo 843   = wceq 1531  wcel 2108  Vcvv 3493  c0 4289  {csn 4559   × cxp 5546  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-dif 3937  df-in 3941  df-ss 3950  df-nul 4290  df-sn 4560  df-rel 5555
This theorem is referenced by: (None)
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