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

Step	Hyp	Ref	Expression
1		dmsnn0 6227	. 2 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2		dm0rn0 5935	. . 3 ⊢ (dom {𝐴} = ∅ ↔ ran {𝐴} = ∅)
3	2	necon3bii 2993	. 2 ⊢ (dom {𝐴} ≠ ∅ ↔ ran {𝐴} ≠ ∅)
4	1, 3	bitri 275	1 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)

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