Theorem dmsnn0 5518

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 3176	. . . . 5 ⊢ 𝑥 ∈ V
2	1	eldm 5243	. . . 4 ⊢ (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦)
3		df-br 4584	. . . . . 6 ⊢ (𝑥{𝐴}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {𝐴})
4		opex 4859	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
5	4	elsn 4140	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {𝐴} ↔ ⟨𝑥, 𝑦⟩ = 𝐴)
6		eqcom 2617	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 ↔ 𝐴 = ⟨𝑥, 𝑦⟩)
7	3, 5, 6	3bitri 285	. . . . 5 ⊢ (𝑥{𝐴}𝑦 ↔ 𝐴 = ⟨𝑥, 𝑦⟩)
8	7	exbii 1764	. . . 4 ⊢ (∃𝑦 𝑥{𝐴}𝑦 ↔ ∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
9	2, 8	bitr2i 264	. . 3 ⊢ (∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝑥 ∈ dom {𝐴})
10	9	exbii 1764	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
11		elvv 5100	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
12		n0 3890	. 2 ⊢ (dom {𝐴} ≠ ∅ ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
13	10, 11, 12	3bitr4i 291	1 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)

Syntax hints:   ↔ wb 195   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583   × cxp 5036  dom cdm 5038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048

This theorem is referenced by:  rnsnn0  5519  dmsn0  5520  dmsn0el  5522  relsn2  5523  1stnpr  7063  1st2val  7085  mpt2xopxnop0  7228  hashfun  13084