Theorem 0nnq 10348

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 5591	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 10336	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3	2	ssrab3 4059	. . 3 ⊢ Q ⊆ (N × N)
4	3	sseli 3965	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
5	1, 4	mto 199	1 ⊢ ¬ ∅ ∈ Q

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114  ∀wral 3140  ∅c0 4293   class class class wbr 5068   × cxp 5555  ‘cfv 6357  2^nd c2nd 7690  Ncnpi 10268   <_N clti 10271   ~_Q ceq 10275  Qcnq 10276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-opab 5131  df-xp 5563  df-nq 10336

This theorem is referenced by:  adderpq  10380  mulerpq  10381  addassnq  10382  mulassnq  10383  distrnq  10385  recmulnq  10388  recclnq  10390  ltanq  10395  ltmnq  10396  ltexnq  10399  nsmallnq  10401  ltbtwnnq  10402  ltrnq  10403  prlem934  10457  ltaddpr  10458  ltexprlem2  10461  ltexprlem3  10462  ltexprlem4  10463  ltexprlem6  10465  ltexprlem7  10466  prlem936  10471  reclem2pr  10472