Theorem dmsnm 5135

Description: The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)

Assertion

Ref	Expression
dmsnm	⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})

Distinct variable group:   𝑥,𝐴

Proof of Theorem dmsnm

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elvv 4725	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 2766	. . . . 5 ⊢ 𝑥 ∈ V
3	2	eldm 4863	. . . 4 ⊢ (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦)
4		df-br 4034	. . . . . 6 ⊢ (𝑥{𝐴}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {𝐴})
5		vex 2766	. . . . . . . 8 ⊢ 𝑦 ∈ V
6	2, 5	opex 4262	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
7	6	elsn 3638	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {𝐴} ↔ ⟨𝑥, 𝑦⟩ = 𝐴)
8		eqcom 2198	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 ↔ 𝐴 = ⟨𝑥, 𝑦⟩)
9	4, 7, 8	3bitri 206	. . . . 5 ⊢ (𝑥{𝐴}𝑦 ↔ 𝐴 = ⟨𝑥, 𝑦⟩)
10	9	exbii 1619	. . . 4 ⊢ (∃𝑦 𝑥{𝐴}𝑦 ↔ ∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
11	3, 10	bitr2i 185	. . 3 ⊢ (∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝑥 ∈ dom {𝐴})
12	11	exbii 1619	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
13	1, 12	bitri 184	1 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})

Syntax hints:   ↔ wb 105   = wceq 1364  ∃wex 1506   ∈ wcel 2167  Vcvv 2763  {csn 3622  ⟨cop 3625   class class class wbr 4033   × cxp 4661  dom cdm 4663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-dm 4673

This theorem is referenced by:  rnsnm  5136  dmsn0  5137  dmsn0el  5139  relsn2m  5140