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Theorem rnsnn0 6164
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 6163 . 2 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2 dm0rn0 5884 . . 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 205  wcel 2107  wne 2940  Vcvv 3447  c0 4286  {csn 4590   × cxp 5635  dom cdm 5637  ran crn 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648
This theorem is referenced by:  2ndnpr  7930  2nd2val  7954
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