Theorem rnsnn0 6196

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 6195	. 2 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2		dm0rn0 5901	. . 3 ⊢ (dom {𝐴} = ∅ ↔ ran {𝐴} = ∅)
3	2	necon3bii 3010	. 2 ⊢ (dom {𝐴} ≠ ∅ ↔ ran {𝐴} ≠ ∅)
4	1, 3	bitri 277	1 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)

Syntax hints:   ↔ wb 208   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455  ∅c0 4286  {csn 4583   × cxp 5646  dom cdm 5648  ran crn 5649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-xp 5654  df-cnv 5656  df-dm 5658  df-rn 5659

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