Theorem 0nnq 9743

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.

Step	Hyp	Ref	Expression
1		0nelxp 5141	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 9731	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3		ssrab2 3685	. . . 4 ⊢ {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))} ⊆ (N × N)
4	2, 3	eqsstri 3633	. . 3 ⊢ Q ⊆ (N × N)
5	4	sseli 3597	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
6	1, 5	mto 188	1 ⊢ ¬ ∅ ∈ Q

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1989  ∀wral 2911  {crab 2915  ∅c0 3913   class class class wbr 4651   × cxp 5110  ‘cfv 5886  2^nd c2nd 7164  Ncnpi 9663   <_N clti 9666   ~_Q ceq 9670  Qcnq 9671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-opab 4711  df-xp 5118  df-nq 9731

This theorem is referenced by:  adderpq  9775  mulerpq  9776  addassnq  9777  mulassnq  9778  distrnq  9780  recmulnq  9783  recclnq  9785  ltanq  9790  ltmnq  9791  ltexnq  9794  nsmallnq  9796  ltbtwnnq  9797  ltrnq  9798  prlem934  9852  ltaddpr  9853  ltexprlem2  9856  ltexprlem3  9857  ltexprlem4  9858  ltexprlem6  9860  ltexprlem7  9861  prlem936  9866  reclem2pr  9867