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Theorem enqex 6822
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 6774 . . . 4 N ∈ V
21, 1xpex 4511 . . 3 (N × N) ∈ V
32, 2xpex 4511 . 2 ((N × N) × (N × N)) ∈ V
4 df-enq 6809 . . 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 4470 . . 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 3040 . 2 ~Q ⊆ ((N × N) × (N × N))
73, 6ssexi 3942 1 ~Q ∈ V
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wex 1422  wcel 1434  Vcvv 2612  cop 3425  {copab 3864   × cxp 4399  (class class class)co 5591  Ncnpi 6734   ·N cmi 6736   ~Q ceq 6741
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-opab 3866  df-iom 4369  df-xp 4407  df-ni 6766  df-enq 6809
This theorem is referenced by:  1nq  6828  addpipqqs  6832  mulpipqqs  6835  ordpipqqs  6836  addclnq  6837  mulclnq  6838  dmaddpq  6841  dmmulpq  6842  recexnq  6852  ltexnqq  6870  prarloclemarch  6880  prarloclemarch2  6881  nnnq  6884  nqpnq0nq  6915  prarloclemlt  6955  prarloclemlo  6956  prarloclemcalc  6964  nqprm  7004
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