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Theorem enqex 6344
Description: The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
Assertion
Ref Expression
enqex ~Q V

Proof of Theorem enqex
Dummy variables x y z w v u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 niex 6296 . . . 4 N V
21, 1xpex 4396 . . 3 (N × N) V
32, 2xpex 4396 . 2 ((N × N) × (N × N)) V
4 df-enq 6331 . . 3 ~Q = {⟨x, y⟩ ∣ ((x (N × N) y (N × N)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z ·N u) = (w ·N v)))}
5 opabssxp 4357 . . 3 {⟨x, y⟩ ∣ ((x (N × N) y (N × N)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z ·N u) = (w ·N v)))} ⊆ ((N × N) × (N × N))
64, 5eqsstri 2969 . 2 ~Q ⊆ ((N × N) × (N × N))
73, 6ssexi 3886 1 ~Q V
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cop 3370  {copab 3808   × cxp 4286  (class class class)co 5455  Ncnpi 6256   ·N cmi 6258   ~Q ceq 6263
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-in1 544  ax-in2 545  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-13 1401  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  ax-un 4136  ax-iinf 4254
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-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-opab 3810  df-iom 4257  df-xp 4294  df-ni 6288  df-enq 6331
This theorem is referenced by:  1nq  6350  addpipqqs  6354  mulpipqqs  6357  ordpipqqs  6358  addclnq  6359  mulclnq  6360  dmaddpq  6363  dmmulpq  6364  recexnq  6374  ltexnqq  6391  prarloclemarch  6401  prarloclemarch2  6402  nnnq  6405  nqpnq0nq  6436  prarloclemlt  6476  prarloclemlo  6477  prarloclemcalc  6485  nqprm  6525
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