Theorem rnsnn0 6184

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 6183	. 2 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2		dm0rn0 5891	. . 3 ⊢ (dom {𝐴} = ∅ ↔ ran {𝐴} = ∅)
3	2	necon3bii 2978	. 2 ⊢ (dom {𝐴} ≠ ∅ ↔ ran {𝐴} ≠ ∅)
4	1, 3	bitri 275	1 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)

Syntax hints:   ↔ wb 206   ∈ wcel 2109   ≠ wne 2926  Vcvv 3450  ∅c0 4299  {csn 4592   × cxp 5639  dom cdm 5641  ran crn 5642

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-cnv 5649  df-dm 5651  df-rn 5652

This theorem is referenced by:  2ndnpr  7976  2nd2val  8000