Theorem ltrelpi 10955

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 10941	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4183	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3977	1 ⊢ <N ⊆ (N × N)

This proof depends on syntax axioms:   ∩ cin 3898   ⊆ wss 3899   E cep 5550   × cxp 5649  Ncnpi 10910   <_N clti 10913

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-in 3906  df-ss 3916  df-lti 10941

This theorem is used by:  ltapi  10969  ltmpi  10970  nlt1pi  10972  indpi  10973  ordpipq  11008  ltsonq  11035  archnq  11046