Theorem ltrelnq 11041

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 11033	. 2 ⊢ <Q = ( <pQ ∩ (Q × Q))
2		inss2 4137	. 2 ⊢ ( <pQ ∩ (Q × Q)) ⊆ (Q × Q)
3	1, 2	eqsstri 3959	1 ⊢ <Q ⊆ (Q × Q)

This proof depends on syntax axioms:   ⊆ wss 3900   ∩ cin 4092   × cxp 5654   <_pQ cltpq 10965  Qcnq 10967   <_Q cltq 10973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-ltnq 11033

This theorem is used by:  lterpq  11085  ltanq  11086  ltmnq  11087  ltexnq  11090  ltbtwnnq  11093  ltrnq  11094  prcdnq  11108  prnmadd  11112  genpcd  11121  nqpr  11129  1idpr  11144  prlem934  11148  ltexprlem4  11154  prlem936  11162  reclem2pr  11163  reclem3pr  11164  reclem4pr  11165