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Theorem enqex 6486
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 6438 . . . 4 N ∈ V
21, 1xpex 4478 . . 3 (N × N) ∈ V
32, 2xpex 4478 . 2 ((N × N) × (N × N)) ∈ V
4 df-enq 6473 . . 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 4439 . . 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 3000 . 2 ~Q ⊆ ((N × N) × (N × N))
73, 6ssexi 3920 1 ~Q ∈ V
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1257  wex 1395  wcel 1407  Vcvv 2572  cop 3403  {copab 3842   × cxp 4368  (class class class)co 5537  Ncnpi 6398   ·N cmi 6400   ~Q ceq 6405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-opab 3844  df-iom 4339  df-xp 4376  df-ni 6430  df-enq 6473
This theorem is referenced by:  1nq  6492  addpipqqs  6496  mulpipqqs  6499  ordpipqqs  6500  addclnq  6501  mulclnq  6502  dmaddpq  6505  dmmulpq  6506  recexnq  6516  ltexnqq  6534  prarloclemarch  6544  prarloclemarch2  6545  nnnq  6548  nqpnq0nq  6579  prarloclemlt  6619  prarloclemlo  6620  prarloclemcalc  6628  nqprm  6668
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