MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rnsnn0 Structured version   Visualization version   GIF version

Theorem rnsnn0 6228
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 6227 . 2 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2 dm0rn0 5935 . . 3 (dom {𝐴} = ∅ ↔ ran {𝐴} = ∅)
32necon3bii 2993 . 2 (dom {𝐴} ≠ ∅ ↔ ran {𝐴} ≠ ∅)
41, 3bitri 275 1 (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2108  wne 2940  Vcvv 3480  c0 4333  {csn 4626   × cxp 5683  dom cdm 5685  ran crn 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  2ndnpr  8019  2nd2val  8043
  Copyright terms: Public domain W3C validator