Theorem ltrelpi 10892

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 10878	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4193	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3986	1 ⊢ <N ⊆ (N × N)

This proof depends on syntax axioms:   ∩ cin 3907   ⊆ wss 3908   E cep 5565   × cxp 5664  Ncnpi 10847   <_N clti 10850

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-in 3915  df-ss 3925  df-lti 10878

This theorem is used by:  ltapi  10906  ltmpi  10907  nlt1pi  10909  indpi  10910  ordpipq  10945  ltsonq  10972  archnq  10983