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Theorem enq0ex 7426
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 4589 . . . 4 ω ∈ V
2 niex 7299 . . . 4 N ∈ V
31, 2xpex 4738 . . 3 (ω × N) ∈ V
43, 3xpex 4738 . 2 ((ω × N) × (ω × N)) ∈ V
5 df-enq0 7411 . . 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 4697 . . 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 3187 . 2 ~Q0 ⊆ ((ω × N) × (ω × N))
84, 7ssexi 4138 1 ~Q0 ∈ V
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wex 1492  wcel 2148  Vcvv 2737  cop 3594  {copab 4060  ωcom 4586   × cxp 4621  (class class class)co 5869   ·o comu 6409  Ncnpi 7259   ~Q0 ceq0 7273
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-opab 4062  df-iom 4587  df-xp 4629  df-ni 7291  df-enq0 7411
This theorem is referenced by:  nqnq0  7428  addnnnq0  7436  mulnnnq0  7437  addclnq0  7438  mulclnq0  7439  prarloclemcalc  7489
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