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Theorem enq0ex 6294
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 4243 . . . 4 𝜔 V
2 niex 6172 . . . 4 N V
31, 2xpex 4380 . . 3 (𝜔 × N) V
43, 3xpex 4380 . 2 ((𝜔 × N) × (𝜔 × N)) V
5 df-enq0 6279 . . 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 4341 . . 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 2952 . 2 ~Q0 ⊆ ((𝜔 × N) × (𝜔 × N))
84, 7ssexi 3869 1 ~Q0 V
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228  wex 1362   wcel 1374  Vcvv 2535  cop 3353  {copab 3791  𝜔com 4240   × cxp 4270  (class class class)co 5436   ·𝑜 comu 5914  Ncnpi 6130   ~Q0 ceq0 6144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-opab 3793  df-iom 4241  df-xp 4278  df-ni 6164  df-enq0 6279
This theorem is referenced by:  nqnq0  6296  addnnnq0  6304  mulnnnq0  6305  addclnq0  6306  mulclnq0  6307  prarloclemcalc  6356
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