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Theorem enqex 6898
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 6850 . . . 4 N ∈ V
21, 1xpex 4541 . . 3 (N × N) ∈ V
32, 2xpex 4541 . 2 ((N × N) × (N × N)) ∈ V
4 df-enq 6885 . . 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 4500 . . 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 3054 . 2 ~Q ⊆ ((N × N) × (N × N))
73, 6ssexi 3969 1 ~Q ∈ V
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  cop 3444  {copab 3890   × cxp 4426  (class class class)co 5634  Ncnpi 6810   ·N cmi 6812   ~Q ceq 6817
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-opab 3892  df-iom 4396  df-xp 4434  df-ni 6842  df-enq 6885
This theorem is referenced by:  1nq  6904  addpipqqs  6908  mulpipqqs  6911  ordpipqqs  6912  addclnq  6913  mulclnq  6914  dmaddpq  6917  dmmulpq  6918  recexnq  6928  ltexnqq  6946  prarloclemarch  6956  prarloclemarch2  6957  nnnq  6960  nqpnq0nq  6991  prarloclemlt  7031  prarloclemlo  7032  prarloclemcalc  7040  nqprm  7080
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