Theorem ltrelpi 10862

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 10848	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4192	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3985	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3906   ⊆ wss 3907   E cep 5551   × cxp 5650  Ncnpi 10817   <_N clti 10820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-in 3914  df-ss 3924  df-lti 10848

This theorem is referenced by:  ltapi  10876  ltmpi  10877  nlt1pi  10879  indpi  10880  ordpipq  10915  ltsonq  10942  archnq  10953