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Theorem dmsnm 5004
 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.
StepHypRef Expression
1 elvv 4601 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 2689 . . . . 5 𝑥 ∈ V
32eldm 4736 . . . 4 (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦)
4 df-br 3930 . . . . . 6 (𝑥{𝐴}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {𝐴})
5 vex 2689 . . . . . . . 8 𝑦 ∈ V
62, 5opex 4151 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
76elsn 3543 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {𝐴} ↔ ⟨𝑥, 𝑦⟩ = 𝐴)
8 eqcom 2141 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴𝐴 = ⟨𝑥, 𝑦⟩)
94, 7, 83bitri 205 . . . . 5 (𝑥{𝐴}𝑦𝐴 = ⟨𝑥, 𝑦⟩)
109exbii 1584 . . . 4 (∃𝑦 𝑥{𝐴}𝑦 ↔ ∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
113, 10bitr2i 184 . . 3 (∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝑥 ∈ dom {𝐴})
1211exbii 1584 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
131, 12bitri 183 1 (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686  {csn 3527  ⟨cop 3530   class class class wbr 3929   × cxp 4537  dom cdm 4539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-dm 4549 This theorem is referenced by:  rnsnm  5005  dmsn0  5006  dmsn0el  5008  relsn2m  5009
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