Theorem ltrelpi 10958

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 10944	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4259	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 4043	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3975   ⊆ wss 3976   E cep 5598   × cxp 5698  Ncnpi 10913   <_N clti 10916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-lti 10944

This theorem is referenced by:  ltapi  10972  ltmpi  10973  nlt1pi  10975  indpi  10976  ordpipq  11011  ltsonq  11038  archnq  11049