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Theorem elvv 5619
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 5571 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
2 vex 3495 . . . . 5 𝑥 ∈ V
3 vex 3495 . . . . 5 𝑦 ∈ V
42, 3pm3.2i 471 . . . 4 (𝑥 ∈ V ∧ 𝑦 ∈ V)
54biantru 530 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
652exbii 1840 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
71, 6bitr4i 279 1 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492  cop 4563   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120  df-xp 5554
This theorem is referenced by:  elvvv  5620  elvvuni  5621  elopaelxp  5634  elrel  5664  copsex2gb  5672  relop  5714  elreldm  5798  dmsnn0  6057  funsndifnop  6905  1stval2  7695  2ndval2  7696  1st2val  7706  2nd2val  7707  dfopab2  7739  dfoprab3s  7740  dftpos4  7900  tpostpos  7901  fundmen  8571  fundmge2nop0  13838  ssrelf  30294  dfdm5  32913  dfrn5  32914  brtxp2  33239  pprodss4v  33242  brpprod3a  33244  brimg  33295  brxrn2  35507  fun2dmnopgexmpl  43360
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