Theorem ltrelnq 9786

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 9778	. 2 ⊢ <Q = ( <pQ ∩ (Q × Q))
2		inss2 3867	. 2 ⊢ ( <pQ ∩ (Q × Q)) ⊆ (Q × Q)
3	1, 2	eqsstri 3668	1 ⊢ <Q ⊆ (Q × Q)

Syntax hints:   ∩ cin 3606   ⊆ wss 3607   × cxp 5141   <_pQ cltpq 9710  Qcnq 9712   <_Q cltq 9718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621  df-ltnq 9778

This theorem is referenced by:  lterpq  9830  ltanq  9831  ltmnq  9832  ltexnq  9835  ltbtwnnq  9838  ltrnq  9839  prcdnq  9853  prnmadd  9857  genpcd  9866  nqpr  9874  1idpr  9889  prlem934  9893  ltexprlem4  9899  prlem936  9907  reclem2pr  9908  reclem3pr  9909  reclem4pr  9910