Theorem ltrelpi 10844

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 10830	. 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 3903   ⊆ wss 3904   E cep 5544   × cxp 5643  Ncnpi 10799   <_N clti 10802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-in 3911  df-ss 3921  df-lti 10830

This theorem is referenced by:  ltapi  10858  ltmpi  10859  nlt1pi  10861  indpi  10862  ordpipq  10897  ltsonq  10924  archnq  10935