Theorem ltrelpi 10576

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 10562	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4160	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3951	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3882   ⊆ wss 3883   E cep 5485   × cxp 5578  Ncnpi 10531   <_N clti 10534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900  df-lti 10562

This theorem is referenced by:  ltapi  10590  ltmpi  10591  nlt1pi  10593  indpi  10594  ordpipq  10629  ltsonq  10656  archnq  10667