Theorem ltrelpi 10645

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 10631	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4163	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3955	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3886   ⊆ wss 3887   E cep 5494   × cxp 5587  Ncnpi 10600   <_N clti 10603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-lti 10631

This theorem is referenced by:  ltapi  10659  ltmpi  10660  nlt1pi  10662  indpi  10663  ordpipq  10698  ltsonq  10725  archnq  10736