Theorem ltrelpi 10747

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 10733	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4177	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3966	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3897   ⊆ wss 3898   E cep 5524   × cxp 5619  Ncnpi 10702   <_N clti 10705

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-in 3905  df-ss 3915  df-lti 10733

This theorem is referenced by:  ltapi  10761  ltmpi  10762  nlt1pi  10764  indpi  10765  ordpipq  10800  ltsonq  10827  archnq  10838