Theorem ltrelpi 10875

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 10861	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4191	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3984	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3905   ⊆ wss 3906   E cep 5562   × cxp 5661  Ncnpi 10830   <_N clti 10833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-in 3913  df-ss 3923  df-lti 10861

This theorem is referenced by:  ltapi  10889  ltmpi  10890  nlt1pi  10892  indpi  10893  ordpipq  10928  ltsonq  10955  archnq  10966