Theorem ltrelpi 10787

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 10773	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4187	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3977	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3897   ⊆ wss 3898   E cep 5518   × cxp 5617  Ncnpi 10742   <_N clti 10745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-in 3905  df-ss 3915  df-lti 10773

This theorem is referenced by:  ltapi  10801  ltmpi  10802  nlt1pi  10804  indpi  10805  ordpipq  10840  ltsonq  10867  archnq  10878