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Theorem elvv 4345
 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 4305 . 2 (A (V × V) ↔ xy(A = ⟨x, y (x V y V)))
2 vex 2554 . . . . 5 x V
3 vex 2554 . . . . 5 y V
42, 3pm3.2i 257 . . . 4 (x V y V)
54biantru 286 . . 3 (A = ⟨x, y⟩ ↔ (A = ⟨x, y (x V y V)))
652exbii 1494 . 2 (xy A = ⟨x, y⟩ ↔ xy(A = ⟨x, y (x V y V)))
71, 6bitr4i 176 1 (A (V × V) ↔ xy A = ⟨x, y⟩)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370   × cxp 4286 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294 This theorem is referenced by:  elvvv  4346  elvvuni  4347  ssrel  4371  elrel  4385  relop  4429  elreldm  4503  dmsnm  4729  1stval2  5724  2ndval2  5725  dfopab2  5757  dfoprab3s  5758  dftpos4  5819  tpostpos  5820  fundmen  6222
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