Theorem ltrelnq 10837

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 10829	. 2 ⊢ <Q = ( <pQ ∩ (Q × Q))
2		inss2 4190	. 2 ⊢ ( <pQ ∩ (Q × Q)) ⊆ (Q × Q)
3	1, 2	eqsstri 3980	1 ⊢ <Q ⊆ (Q × Q)

Syntax hints:   ∩ cin 3900   ⊆ wss 3901   × cxp 5622   <_pQ cltpq 10761  Qcnq 10763   <_Q cltq 10769

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-in 3908  df-ss 3918  df-ltnq 10829

This theorem is referenced by:  lterpq  10881  ltanq  10882  ltmnq  10883  ltexnq  10886  ltbtwnnq  10889  ltrnq  10890  prcdnq  10904  prnmadd  10908  genpcd  10917  nqpr  10925  1idpr  10940  prlem934  10944  ltexprlem4  10950  prlem936  10958  reclem2pr  10959  reclem3pr  10960  reclem4pr  10961