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Theorem dmsnn0 5598
Description: The domain of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmsnn0 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)

Proof of Theorem dmsnn0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3201 . . . . 5 𝑥 ∈ V
21eldm 5319 . . . 4 (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦)
3 df-br 4652 . . . . . 6 (𝑥{𝐴}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {𝐴})
4 opex 4930 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
54elsn 4190 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {𝐴} ↔ ⟨𝑥, 𝑦⟩ = 𝐴)
6 eqcom 2628 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴𝐴 = ⟨𝑥, 𝑦⟩)
73, 5, 63bitri 286 . . . . 5 (𝑥{𝐴}𝑦𝐴 = ⟨𝑥, 𝑦⟩)
87exbii 1773 . . . 4 (∃𝑦 𝑥{𝐴}𝑦 ↔ ∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
92, 8bitr2i 265 . . 3 (∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝑥 ∈ dom {𝐴})
109exbii 1773 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
11 elvv 5175 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
12 n0 3929 . 2 (dom {𝐴} ≠ ∅ ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
1310, 11, 123bitr4i 292 1 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1482  wex 1703  wcel 1989  wne 2793  Vcvv 3198  c0 3913  {csn 4175  cop 4181   class class class wbr 4651   × cxp 5110  dom cdm 5112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-dm 5122
This theorem is referenced by:  rnsnn0  5599  dmsn0  5600  dmsn0el  5602  relsn2  5603  1stnpr  7169  1st2val  7191  mpt2xopxnop0  7338  hashfun  13219
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