Theorem ltrelnq 10899

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 10891	. 2 ⊢ <Q = ( <pQ ∩ (Q × Q))
2		inss2 4192	. 2 ⊢ ( <pQ ∩ (Q × Q)) ⊆ (Q × Q)
3	1, 2	eqsstri 3985	1 ⊢ <Q ⊆ (Q × Q)

Syntax hints:   ∩ cin 3906   ⊆ wss 3907   × cxp 5650   <_pQ cltpq 10823  Qcnq 10825   <_Q cltq 10831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-in 3914  df-ss 3924  df-ltnq 10891

This theorem is referenced by:  lterpq  10943  ltanq  10944  ltmnq  10945  ltexnq  10948  ltbtwnnq  10951  ltrnq  10952  prcdnq  10966  prnmadd  10970  genpcd  10979  nqpr  10987  1idpr  11002  prlem934  11006  ltexprlem4  11012  prlem936  11020  reclem2pr  11021  reclem3pr  11022  reclem4pr  11023