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Theorem relsn2 5603
Description: A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013.)
Hypothesis
Ref Expression
relsn2.1 𝐴 ∈ V
Assertion
Ref Expression
relsn2 (Rel {𝐴} ↔ dom {𝐴} ≠ ∅)

Proof of Theorem relsn2
StepHypRef Expression
1 relsn2.1 . . 3 𝐴 ∈ V
21relsn 5221 . 2 (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
3 dmsnn0 5598 . 2 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
42, 3bitri 264 1 (Rel {𝐴} ↔ dom {𝐴} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 1989  wne 2793  Vcvv 3198  c0 3913  {csn 4175   × cxp 5110  dom cdm 5112  Rel wrel 5117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-dm 5122
This theorem is referenced by: (None)
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