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Theorem enq0ex 6421
Description: The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
Assertion
Ref Expression
enq0ex ~Q0 V

Proof of Theorem enq0ex
Dummy variables x y z w v u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omex 4259 . . . 4 𝜔 V
2 niex 6296 . . . 4 N V
31, 2xpex 4396 . . 3 (𝜔 × N) V
43, 3xpex 4396 . 2 ((𝜔 × N) × (𝜔 × N)) V
5 df-enq0 6406 . . 3 ~Q0 = {⟨v, u⟩ ∣ ((v (𝜔 × N) u (𝜔 × N)) xyzw((v = ⟨x, y u = ⟨z, w⟩) (x ·𝑜 w) = (y ·𝑜 z)))}
6 opabssxp 4357 . . 3 {⟨v, u⟩ ∣ ((v (𝜔 × N) u (𝜔 × N)) xyzw((v = ⟨x, y u = ⟨z, w⟩) (x ·𝑜 w) = (y ·𝑜 z)))} ⊆ ((𝜔 × N) × (𝜔 × N))
75, 6eqsstri 2969 . 2 ~Q0 ⊆ ((𝜔 × N) × (𝜔 × N))
84, 7ssexi 3886 1 ~Q0 V
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cop 3370  {copab 3808  𝜔com 4256   × cxp 4286  (class class class)co 5455   ·𝑜 comu 5938  Ncnpi 6256   ~Q0 ceq0 6270
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-enq0 6406
This theorem is referenced by:  nqnq0  6423  addnnnq0  6431  mulnnnq0  6432  addclnq0  6433  mulclnq0  6434  prarloclemcalc  6484
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