Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  enq0ex GIF version

Theorem enq0ex 7267
 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 4511 . . . 4 ω ∈ V
2 niex 7140 . . . 4 N ∈ V
31, 2xpex 4658 . . 3 (ω × N) ∈ V
43, 3xpex 4658 . 2 ((ω × N) × (ω × N)) ∈ V
5 df-enq0 7252 . . 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 4617 . . 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 3130 . 2 ~Q0 ⊆ ((ω × N) × (ω × N))
84, 7ssexi 4070 1 ~Q0 ∈ V
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 1481  Vcvv 2687  ⟨cop 3531  {copab 3992  ωcom 4508   × cxp 4541  (class class class)co 5778   ·o comu 6315  Ncnpi 7100   ~Q0 ceq0 7114 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-iinf 4506 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-opab 3994  df-iom 4509  df-xp 4549  df-ni 7132  df-enq0 7252 This theorem is referenced by:  nqnq0  7269  addnnnq0  7277  mulnnnq0  7278  addclnq0  7279  mulclnq0  7280  prarloclemcalc  7330
 Copyright terms: Public domain W3C validator