Theorem ltrelnq 10951

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 10943	. 2 ⊢ <Q = ( <pQ ∩ (Q × Q))
2		inss2 4228	. 2 ⊢ ( <pQ ∩ (Q × Q)) ⊆ (Q × Q)
3	1, 2	eqsstri 4011	1 ⊢ <Q ⊆ (Q × Q)

Syntax hints:   ∩ cin 3943   ⊆ wss 3944   × cxp 5676   <_pQ cltpq 10875  Qcnq 10877   <_Q cltq 10883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-in 3951  df-ss 3961  df-ltnq 10943

This theorem is referenced by:  lterpq  10995  ltanq  10996  ltmnq  10997  ltexnq  11000  ltbtwnnq  11003  ltrnq  11004  prcdnq  11018  prnmadd  11022  genpcd  11031  nqpr  11039  1idpr  11054  prlem934  11058  ltexprlem4  11064  prlem936  11072  reclem2pr  11073  reclem3pr  11074  reclem4pr  11075