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Theorem dmsnm 5076
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 4673 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 2733 . . . . 5 𝑥 ∈ V
32eldm 4808 . . . 4 (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦)
4 df-br 3990 . . . . . 6 (𝑥{𝐴}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {𝐴})
5 vex 2733 . . . . . . . 8 𝑦 ∈ V
62, 5opex 4214 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
76elsn 3599 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {𝐴} ↔ ⟨𝑥, 𝑦⟩ = 𝐴)
8 eqcom 2172 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴𝐴 = ⟨𝑥, 𝑦⟩)
94, 7, 83bitri 205 . . . . 5 (𝑥{𝐴}𝑦𝐴 = ⟨𝑥, 𝑦⟩)
109exbii 1598 . . . 4 (∃𝑦 𝑥{𝐴}𝑦 ↔ ∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
113, 10bitr2i 184 . . 3 (∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝑥 ∈ dom {𝐴})
1211exbii 1598 . 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 1348  wex 1485  wcel 2141  Vcvv 2730  {csn 3583  cop 3586   class class class wbr 3989   × cxp 4609  dom cdm 4611
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-dm 4621
This theorem is referenced by:  rnsnm  5077  dmsn0  5078  dmsn0el  5080  relsn2m  5081
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