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Theorem elvv 5138
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 5091 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
2 vex 3189 . . . . 5 𝑥 ∈ V
3 vex 3189 . . . . 5 𝑦 ∈ V
42, 3pm3.2i 471 . . . 4 (𝑥 ∈ V ∧ 𝑦 ∈ V)
54biantru 526 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
652exbii 1772 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
71, 6bitr4i 267 1 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  cop 4154   × cxp 5072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-opab 4674  df-xp 5080
This theorem is referenced by:  elvvv  5139  elvvuni  5140  elopaelxp  5152  ssrelOLD  5169  elrel  5183  copsex2gb  5191  relop  5232  elreldm  5310  dmsnn0  5559  funsndifnop  6370  1stval2  7130  2ndval2  7131  1st2val  7139  2nd2val  7140  dfopab2  7167  dfoprab3s  7168  dftpos4  7316  tpostpos  7317  fundmen  7974  fundmge2nop0  13213  ssrelf  29265  dfdm5  31375  dfrn5  31376  brtxp2  31627  pprodss4v  31630  brpprod3a  31632  brimg  31683  fun2dmnopgexmpl  40597
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