Theorem ltrelpi 10783

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 10769	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4189	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3982	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3902   ⊆ wss 3903   E cep 5518   × cxp 5617  Ncnpi 10738   <_N clti 10741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-in 3910  df-ss 3920  df-lti 10769

This theorem is referenced by:  ltapi  10797  ltmpi  10798  nlt1pi  10800  indpi  10801  ordpipq  10836  ltsonq  10863  archnq  10874