Theorem dmsnn0 6050

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 3402	. . . . 5 ⊢ 𝑥 ∈ V
2	1	eldm 5754	. . . 4 ⊢ (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦)
3		df-br 5040	. . . . . 6 ⊢ (𝑥{𝐴}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {𝐴})
4		opex 5333	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
5	4	elsn 4542	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {𝐴} ↔ ⟨𝑥, 𝑦⟩ = 𝐴)
6		eqcom 2743	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 ↔ 𝐴 = ⟨𝑥, 𝑦⟩)
7	3, 5, 6	3bitri 300	. . . . 5 ⊢ (𝑥{𝐴}𝑦 ↔ 𝐴 = ⟨𝑥, 𝑦⟩)
8	7	exbii 1855	. . . 4 ⊢ (∃𝑦 𝑥{𝐴}𝑦 ↔ ∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
9	2, 8	bitr2i 279	. . 3 ⊢ (∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝑥 ∈ dom {𝐴})
10	9	exbii 1855	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
11		elvv 5608	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
12		n0 4247	. 2 ⊢ (dom {𝐴} ≠ ∅ ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
13	10, 11, 12	3bitr4i 306	1 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)

Syntax hints:   ↔ wb 209   = wceq 1543  ∃wex 1787   ∈ wcel 2112   ≠ wne 2932  Vcvv 3398  ∅c0 4223  {csn 4527  ⟨cop 4533   class class class wbr 5039   × cxp 5534  dom cdm 5536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ne 2933  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-xp 5542  df-dm 5546

This theorem is referenced by:  rnsnn0  6051  dmsn0  6052  dmsn0el  6054  relsn2  6055  1stnpr  7743  1st2val  7767  mpoxopxnop0  7935  cnvfi  8834  hashfun  13969  fineqvac  32733