Theorem ltrelpi 9590

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 9576	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 3796	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3598	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3539   ⊆ wss 3540   E cep 4947   × cxp 5036  Ncnpi 9545   <_N clti 9548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-lti 9576

This theorem is referenced by:  ltapi  9604  ltmpi  9605  nlt1pi  9607  indpi  9608  ordpipq  9643  ltsonq  9670  archnq  9681