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Theorem enqex 7136
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 7088 . . . 4 N ∈ V
21, 1xpex 4624 . . 3 (N × N) ∈ V
32, 2xpex 4624 . 2 ((N × N) × (N × N)) ∈ V
4 df-enq 7123 . . 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 4583 . . 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 3099 . 2 ~Q ⊆ ((N × N) × (N × N))
73, 6ssexi 4036 1 ~Q ∈ V
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1316  wex 1453  wcel 1465  Vcvv 2660  cop 3500  {copab 3958   × cxp 4507  (class class class)co 5742  Ncnpi 7048   ·N cmi 7050   ~Q ceq 7055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-opab 3960  df-iom 4475  df-xp 4515  df-ni 7080  df-enq 7123
This theorem is referenced by:  1nq  7142  addpipqqs  7146  mulpipqqs  7149  ordpipqqs  7150  addclnq  7151  mulclnq  7152  dmaddpq  7155  dmmulpq  7156  recexnq  7166  ltexnqq  7184  prarloclemarch  7194  prarloclemarch2  7195  nnnq  7198  nqpnq0nq  7229  prarloclemlt  7269  prarloclemlo  7270  prarloclemcalc  7278  nqprm  7318
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