Theorem ltrelpi 10903

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 10889	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4213	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 4005	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3925   ⊆ wss 3926   E cep 5552   × cxp 5652  Ncnpi 10858   <_N clti 10861

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-in 3933  df-ss 3943  df-lti 10889

This theorem is referenced by:  ltapi  10917  ltmpi  10918  nlt1pi  10920  indpi  10921  ordpipq  10956  ltsonq  10983  archnq  10994