MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmsnn0 Structured version   Visualization version   GIF version

Theorem dmsnn0 6163
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 3451 . . . . 5 𝑥 ∈ V
21eldm 5860 . . . 4 (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦)
3 df-br 5110 . . . . . 6 (𝑥{𝐴}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {𝐴})
4 opex 5425 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
54elsn 4605 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {𝐴} ↔ ⟨𝑥, 𝑦⟩ = 𝐴)
6 eqcom 2740 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴𝐴 = ⟨𝑥, 𝑦⟩)
73, 5, 63bitri 297 . . . . 5 (𝑥{𝐴}𝑦𝐴 = ⟨𝑥, 𝑦⟩)
87exbii 1851 . . . 4 (∃𝑦 𝑥{𝐴}𝑦 ↔ ∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
92, 8bitr2i 276 . . 3 (∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝑥 ∈ dom {𝐴})
109exbii 1851 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
11 elvv 5710 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
12 n0 4310 . 2 (dom {𝐴} ≠ ∅ ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
1310, 11, 123bitr4i 303 1 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wex 1782  wcel 2107  wne 2940  Vcvv 3447  c0 4286  {csn 4590  cop 4596   class class class wbr 5109   × cxp 5635  dom cdm 5637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-dm 5647
This theorem is referenced by:  rnsnn0  6164  dmsn0  6165  dmsn0el  6167  relsn2  6168  1stnpr  7929  1st2val  7953  mpoxopxnop0  8150  cnvfi  9130  hashfun  14346  fineqvac  33762
  Copyright terms: Public domain W3C validator