Theorem ltrelnq 10995

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 10987	. 2 ⊢ <Q = ( <pQ ∩ (Q × Q))
2		inss2 4259	. 2 ⊢ ( <pQ ∩ (Q × Q)) ⊆ (Q × Q)
3	1, 2	eqsstri 4043	1 ⊢ <Q ⊆ (Q × Q)

Syntax hints:   ∩ cin 3975   ⊆ wss 3976   × cxp 5698   <_pQ cltpq 10919  Qcnq 10921   <_Q cltq 10927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-ltnq 10987

This theorem is referenced by:  lterpq  11039  ltanq  11040  ltmnq  11041  ltexnq  11044  ltbtwnnq  11047  ltrnq  11048  prcdnq  11062  prnmadd  11066  genpcd  11075  nqpr  11083  1idpr  11098  prlem934  11102  ltexprlem4  11108  prlem936  11116  reclem2pr  11117  reclem3pr  11118  reclem4pr  11119