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

Step	Hyp	Ref	Expression
1		dmsnn0 6163	. 2 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2		dm0rn0 5871	. . 3 ⊢ (dom {𝐴} = ∅ ↔ ran {𝐴} = ∅)
3	2	necon3bii 2982	. 2 ⊢ (dom {𝐴} ≠ ∅ ↔ ran {𝐴} ≠ ∅)
4	1, 3	bitri 275	1 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)

Syntax hints:   ↔ wb 206   ∈ wcel 2113   ≠ wne 2930  Vcvv 3438  ∅c0 4283  {csn 4578   × cxp 5620  dom cdm 5622  ran crn 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-xp 5628  df-cnv 5630  df-dm 5632  df-rn 5633

This theorem is referenced by:  2ndnpr  7936  2nd2val  7960