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Theorem rnsnn0 5589
 Description: The range of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.)
Assertion
Ref Expression
rnsnn0 (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)

Proof of Theorem rnsnn0
StepHypRef Expression
1 dmsnn0 5588 . 2 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2 dm0rn0 5331 . . 3 (dom {𝐴} = ∅ ↔ ran {𝐴} = ∅)
32necon3bii 2843 . 2 (dom {𝐴} ≠ ∅ ↔ ran {𝐴} ≠ ∅)
41, 3bitri 264 1 (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∈ wcel 1988   ≠ wne 2791  Vcvv 3195  ∅c0 3907  {csn 4168   × cxp 5102  dom cdm 5104  ran crn 5105 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-cnv 5112  df-dm 5114  df-rn 5115 This theorem is referenced by:  2ndnpr  7158  2nd2val  7180
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