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

Theorem 0nnq 9602
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 5057 . 2 ¬ ∅ ∈ (N × N)
2 df-nq 9590 . . . 4 Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd𝑥) <N (2nd𝑦))}
3 ssrab2 3649 . . . 4 {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd𝑥) <N (2nd𝑦))} ⊆ (N × N)
42, 3eqsstri 3597 . . 3 Q ⊆ (N × N)
54sseli 3563 . 2 (∅ ∈ Q → ∅ ∈ (N × N))
61, 5mto 186 1 ¬ ∅ ∈ Q
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 1976  wral 2895  {crab 2899  c0 3873   class class class wbr 4577   × cxp 5026  cfv 5790  2nd c2nd 7035  Ncnpi 9522   <N clti 9525   ~Q ceq 9529  Qcnq 9530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-opab 4638  df-xp 5034  df-nq 9590
This theorem is referenced by:  adderpq  9634  mulerpq  9635  addassnq  9636  mulassnq  9637  distrnq  9639  recmulnq  9642  recclnq  9644  ltanq  9649  ltmnq  9650  ltexnq  9653  nsmallnq  9655  ltbtwnnq  9656  ltrnq  9657  prlem934  9711  ltaddpr  9712  ltexprlem2  9715  ltexprlem3  9716  ltexprlem4  9717  ltexprlem6  9719  ltexprlem7  9720  prlem936  9725  reclem2pr  9726
  Copyright terms: Public domain W3C validator