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Theorem elvv 5380
Description: Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
elvv (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem elvv
StepHypRef Expression
1 elxp 5335 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
2 vex 3388 . . . . 5 𝑥 ∈ V
3 vex 3388 . . . . 5 𝑦 ∈ V
42, 3pm3.2i 463 . . . 4 (𝑥 ∈ V ∧ 𝑦 ∈ V)
54biantru 526 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
652exbii 1945 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
71, 6bitr4i 270 1 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wex 1875  wcel 2157  Vcvv 3385  cop 4374   × cxp 5310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-opab 4906  df-xp 5318
This theorem is referenced by:  elvvv  5381  elvvuni  5382  elopaelxp  5396  elrel  5426  copsex2gb  5433  relop  5476  elreldm  5553  dmsnn0  5816  funsndifnop  6644  1stval2  7418  2ndval2  7419  1st2val  7429  2nd2val  7430  dfopab2  7457  dfoprab3s  7458  dftpos4  7609  tpostpos  7610  fundmen  8269  fundmge2nop0  13523  ssrelf  29944  dfdm5  32188  dfrn5  32189  brtxp2  32501  pprodss4v  32504  brpprod3a  32506  brimg  32557  brxrn2  34631  fun2dmnopgexmpl  42139
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