Theorem ltrelpi 10777

Description: Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
ltrelpi	⊢ <N ⊆ (N × N)

Proof of Theorem ltrelpi

Step	Hyp	Ref	Expression
1		df-lti 10763	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4188	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3981	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3901   ⊆ wss 3902   E cep 5515   × cxp 5614  Ncnpi 10732   <_N clti 10735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-in 3909  df-ss 3919  df-lti 10763

This theorem is referenced by:  ltapi  10791  ltmpi  10792  nlt1pi  10794  indpi  10795  ordpipq  10830  ltsonq  10857  archnq  10868