Theorem dmsnn0 6227

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.

Step	Hyp	Ref	Expression
1		vex 3484	. . . . 5 ⊢ 𝑥 ∈ V
2	1	eldm 5911	. . . 4 ⊢ (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦)
3		df-br 5144	. . . . . 6 ⊢ (𝑥{𝐴}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {𝐴})
4		opex 5469	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
5	4	elsn 4641	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {𝐴} ↔ ⟨𝑥, 𝑦⟩ = 𝐴)
6		eqcom 2744	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 ↔ 𝐴 = ⟨𝑥, 𝑦⟩)
7	3, 5, 6	3bitri 297	. . . . 5 ⊢ (𝑥{𝐴}𝑦 ↔ 𝐴 = ⟨𝑥, 𝑦⟩)
8	7	exbii 1848	. . . 4 ⊢ (∃𝑦 𝑥{𝐴}𝑦 ↔ ∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
9	2, 8	bitr2i 276	. . 3 ⊢ (∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝑥 ∈ dom {𝐴})
10	9	exbii 1848	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
11		elvv 5760	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
12		n0 4353	. 2 ⊢ (dom {𝐴} ≠ ∅ ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
13	10, 11, 12	3bitr4i 303	1 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)

Syntax hints:   ↔ wb 206   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  Vcvv 3480  ∅c0 4333  {csn 4626  ⟨cop 4632   class class class wbr 5143   × cxp 5683  dom cdm 5685

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-dm 5695

This theorem is referenced by:  rnsnn0  6228  dmsn0  6229  dmsn0el  6231  relsn2  6232  1stnpr  8018  1st2val  8042  mpoxopxnop0  8240  cnvfi  9216  hashfun  14476  fineqvac  35111