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Theorem enq0ex 6680
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 4336 . . . 4 ω ∈ V
2 niex 6553 . . . 4 N ∈ V
31, 2xpex 4475 . . 3 (ω × N) ∈ V
43, 3xpex 4475 . 2 ((ω × N) × (ω × N)) ∈ V
5 df-enq0 6665 . . 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 4434 . . 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 3030 . 2 ~Q0 ⊆ ((ω × N) × (ω × N))
84, 7ssexi 3918 1 ~Q0 ∈ V
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wex 1422  wcel 1434  Vcvv 2602  cop 3403  {copab 3840  ωcom 4333   × cxp 4363  (class class class)co 5537   ·𝑜 comu 6057  Ncnpi 6513   ~Q0 ceq0 6527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-iinf 4331
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-opab 3842  df-iom 4334  df-xp 4371  df-ni 6545  df-enq0 6665
This theorem is referenced by:  nqnq0  6682  addnnnq0  6690  mulnnnq0  6691  addclnq0  6692  mulclnq0  6693  prarloclemcalc  6743
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