Theorem 0nnq 9746

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	|- -. (/) e. Q.

Proof of Theorem 0nnq

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5143	. 2 |- -. (/) e. ( N. X. N. )
2		df-nq 9734	. . . 4 |- Q. = { y e. ( N. X. N. ) | A. x e. ( N. X. N. ) ( y ~Q x -> -. ( 2nd ` x ) <N ( 2nd ` y ) ) }
3		ssrab2 3687	. . . 4 |- { y e. ( N. X. N. ) | A. x e. ( N. X. N. ) ( y ~Q x -> -. ( 2nd ` x ) <N ( 2nd ` y ) ) } C_ ( N. X. N. )
4	2, 3	eqsstri 3635	. . 3 |- Q. C_ ( N. X. N. )
5	4	sseli 3599	. 2 |- ( (/) e. Q. -> (/) e. ( N. X. N. ) )
6	1, 5	mto 188	1 |- -. (/) e. Q.

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1990   A.wral 2912   {crab 2916   (/)c0 3915   class class class wbr 4653   X. cxp 5112   ` cfv 5888   2ndc2nd 7167   N.cnpi 9666   <N clti 9669   ~Q ceq 9673   Q.cnq 9674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-nq 9734

This theorem is referenced by:  adderpq  9778  mulerpq  9779  addassnq  9780  mulassnq  9781  distrnq  9783  recmulnq  9786  recclnq  9788  ltanq  9793  ltmnq  9794  ltexnq  9797  nsmallnq  9799  ltbtwnnq  9800  ltrnq  9801  prlem934  9855  ltaddpr  9856  ltexprlem2  9859  ltexprlem3  9860  ltexprlem4  9861  ltexprlem6  9863  ltexprlem7  9864  prlem936  9869  reclem2pr  9870