Theorem ltrelnq 10849

Description: Positive fraction 'less than' is a relation on positive fractions. (Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltrelnq	⊢ <Q ⊆ (Q × Q)

Proof of Theorem ltrelnq

Step	Hyp	Ref	Expression
1		df-ltnq 10841	. 2 ⊢ <Q = ( <pQ ∩ (Q × Q))
2		inss2 4192	. 2 ⊢ ( <pQ ∩ (Q × Q)) ⊆ (Q × Q)
3	1, 2	eqsstri 3982	1 ⊢ <Q ⊆ (Q × Q)

Syntax hints:   ∩ cin 3902   ⊆ wss 3903   × cxp 5630   <_pQ cltpq 10773  Qcnq 10775   <_Q cltq 10781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-in 3910  df-ss 3920  df-ltnq 10841

This theorem is referenced by:  lterpq  10893  ltanq  10894  ltmnq  10895  ltexnq  10898  ltbtwnnq  10901  ltrnq  10902  prcdnq  10916  prnmadd  10920  genpcd  10929  nqpr  10937  1idpr  10952  prlem934  10956  ltexprlem4  10962  prlem936  10970  reclem2pr  10971  reclem3pr  10972  reclem4pr  10973