MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0nnq Structured version   Visualization version   GIF version

Theorem 0nnq 9784
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 5177 . 2 ¬ ∅ ∈ (N × N)
2 df-nq 9772 . . . 4 Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd𝑥) <N (2nd𝑦))}
3 ssrab2 3720 . . . 4 {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd𝑥) <N (2nd𝑦))} ⊆ (N × N)
42, 3eqsstri 3668 . . 3 Q ⊆ (N × N)
54sseli 3632 . 2 (∅ ∈ Q → ∅ ∈ (N × N))
61, 5mto 188 1 ¬ ∅ ∈ Q
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2030  wral 2941  {crab 2945  c0 3948   class class class wbr 4685   × cxp 5141  cfv 5926  2nd c2nd 7209  Ncnpi 9704   <N clti 9707   ~Q ceq 9711  Qcnq 9712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-nq 9772
This theorem is referenced by:  adderpq  9816  mulerpq  9817  addassnq  9818  mulassnq  9819  distrnq  9821  recmulnq  9824  recclnq  9826  ltanq  9831  ltmnq  9832  ltexnq  9835  nsmallnq  9837  ltbtwnnq  9838  ltrnq  9839  prlem934  9893  ltaddpr  9894  ltexprlem2  9897  ltexprlem3  9898  ltexprlem4  9899  ltexprlem6  9901  ltexprlem7  9902  prlem936  9907  reclem2pr  9908
  Copyright terms: Public domain W3C validator