Theorem ltrelpi 10927

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 10913	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4246	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 4030	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3962   ⊆ wss 3963   E cep 5588   × cxp 5687  Ncnpi 10882   <_N clti 10885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-lti 10913

This theorem is referenced by:  ltapi  10941  ltmpi  10942  nlt1pi  10944  indpi  10945  ordpipq  10980  ltsonq  11007  archnq  11018