Theorem ltrelpi 9749

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 9735	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 3867	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3668	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3606   ⊆ wss 3607   E cep 5057   × cxp 5141  Ncnpi 9704   <_N clti 9707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621  df-lti 9735

This theorem is referenced by:  ltapi  9763  ltmpi  9764  nlt1pi  9766  indpi  9767  ordpipq  9802  ltsonq  9829  archnq  9840