Theorem rnsnn0 6198

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 6197	. 2 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2		dm0rn0 5902	. . 3 ⊢ (dom {𝐴} = ∅ ↔ ran {𝐴} = ∅)
3	2	necon3bii 3007	. 2 ⊢ (dom {𝐴} ≠ ∅ ↔ ran {𝐴} ≠ ∅)
4	1, 3	bitri 278	1 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2145   ≠ wne 2955  Vcvv 3450  ∅c0 4278  {csn 4583   × cxp 5645  dom cdm 5647  ran crn 5648

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-cnv 5655  df-dm 5657  df-rn 5658

This theorem is used by:  2ndnpr  7989  2nd2val  8013