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Theorem relsnb 5464
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 5463 . . . 4 (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
21biimpcd 241 . . 3 (Rel {𝐴} → (𝐴 ∈ V → 𝐴 ∈ (V × V)))
3 imor 884 . . 3 ((𝐴 ∈ V → 𝐴 ∈ (V × V)) ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
42, 3sylib 210 . 2 (Rel {𝐴} → (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
5 snprc 4473 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
6 rel0 5461 . . . . 5 Rel ∅
7 releq 5440 . . . . 5 ({𝐴} = ∅ → (Rel {𝐴} ↔ Rel ∅))
86, 7mpbiri 250 . . . 4 ({𝐴} = ∅ → Rel {𝐴})
95, 8sylbi 209 . . 3 𝐴 ∈ V → Rel {𝐴})
10 relsng 5463 . . . 4 (𝐴 ∈ (V × V) → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
1110ibir 260 . . 3 (𝐴 ∈ (V × V) → Rel {𝐴})
129, 11jaoi 888 . 2 ((¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)) → Rel {𝐴})
134, 12impbii 201 1 (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wo 878   = wceq 1656  wcel 2164  Vcvv 3414  c0 4146  {csn 4399   × cxp 5344  Rel wrel 5351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-dif 3801  df-in 3805  df-ss 3812  df-nul 4147  df-sn 4400  df-rel 5353
This theorem is referenced by: (None)
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