Theorem ltrelnq 10921

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 10913	. 2 ⊢ <Q = ( <pQ ∩ (Q × Q))
2		inss2 4230	. 2 ⊢ ( <pQ ∩ (Q × Q)) ⊆ (Q × Q)
3	1, 2	eqsstri 4017	1 ⊢ <Q ⊆ (Q × Q)

Syntax hints:   ∩ cin 3948   ⊆ wss 3949   × cxp 5675   <_pQ cltpq 10845  Qcnq 10847   <_Q cltq 10853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-ltnq 10913

This theorem is referenced by:  lterpq  10965  ltanq  10966  ltmnq  10967  ltexnq  10970  ltbtwnnq  10973  ltrnq  10974  prcdnq  10988  prnmadd  10992  genpcd  11001  nqpr  11009  1idpr  11024  prlem934  11028  ltexprlem4  11034  prlem936  11042  reclem2pr  11043  reclem3pr  11044  reclem4pr  11045