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Theorem enq0ex 7469
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 4610 . . . 4 ω ∈ V
2 niex 7342 . . . 4 N ∈ V
31, 2xpex 4759 . . 3 (ω × N) ∈ V
43, 3xpex 4759 . 2 ((ω × N) × (ω × N)) ∈ V
5 df-enq0 7454 . . 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 4718 . . 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 3202 . 2 ~Q0 ⊆ ((ω × N) × (ω × N))
84, 7ssexi 4156 1 ~Q0 ∈ V
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1364  wex 1503  wcel 2160  Vcvv 2752  cop 3610  {copab 4078  ωcom 4607   × cxp 4642  (class class class)co 5897   ·o comu 6440  Ncnpi 7302   ~Q0 ceq0 7316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-opab 4080  df-iom 4608  df-xp 4650  df-ni 7334  df-enq0 7454
This theorem is referenced by:  nqnq0  7471  addnnnq0  7479  mulnnnq0  7480  addclnq0  7481  mulclnq0  7482  prarloclemcalc  7532
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