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Theorem 0nnq 9625
Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
0nnq ¬ ∅ ∈ Q

Proof of Theorem 0nnq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0nelxp 5067 . 2 ¬ ∅ ∈ (N × N)
2 df-nq 9613 . . . 4 Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd𝑥) <N (2nd𝑦))}
3 ssrab2 3650 . . . 4 {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd𝑥) <N (2nd𝑦))} ⊆ (N × N)
42, 3eqsstri 3598 . . 3 Q ⊆ (N × N)
54sseli 3564 . 2 (∅ ∈ Q → ∅ ∈ (N × N))
61, 5mto 187 1 ¬ ∅ ∈ Q
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 1977  wral 2896  {crab 2900  c0 3874   class class class wbr 4583   × cxp 5036  cfv 5804  2nd c2nd 7058  Ncnpi 9545   <N clti 9548   ~Q ceq 9552  Qcnq 9553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-nq 9613
This theorem is referenced by:  adderpq  9657  mulerpq  9658  addassnq  9659  mulassnq  9660  distrnq  9662  recmulnq  9665  recclnq  9667  ltanq  9672  ltmnq  9673  ltexnq  9676  nsmallnq  9678  ltbtwnnq  9679  ltrnq  9680  prlem934  9734  ltaddpr  9735  ltexprlem2  9738  ltexprlem3  9739  ltexprlem4  9740  ltexprlem6  9742  ltexprlem7  9743  prlem936  9748  reclem2pr  9749
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