Theorem ltrelnq 9748

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 9740	. 2 ⊢ <Q = ( <pQ ∩ (Q × Q))
2		inss2 3834	. 2 ⊢ ( <pQ ∩ (Q × Q)) ⊆ (Q × Q)
3	1, 2	eqsstri 3635	1 ⊢ <Q ⊆ (Q × Q)

Syntax hints:   ∩ cin 3573   ⊆ wss 3574   × cxp 5112   <_pQ cltpq 9672  Qcnq 9674   <_Q cltq 9680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-ltnq 9740

This theorem is referenced by:  lterpq  9792  ltanq  9793  ltmnq  9794  ltexnq  9797  ltbtwnnq  9800  ltrnq  9801  prcdnq  9815  prnmadd  9819  genpcd  9828  nqpr  9836  1idpr  9851  prlem934  9855  ltexprlem4  9861  prlem936  9869  reclem2pr  9870  reclem3pr  9871  reclem4pr  9872