Theorem ltrelpi 10812

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 10798	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4192	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3982	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3902   ⊆ wss 3903   E cep 5531   × cxp 5630  Ncnpi 10767   <_N clti 10770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-in 3910  df-ss 3920  df-lti 10798

This theorem is referenced by:  ltapi  10826  ltmpi  10827  nlt1pi  10829  indpi  10830  ordpipq  10865  ltsonq  10892  archnq  10903