Theorem ltrelpi 10977

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 10963	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4134	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3954	1 ⊢ <N ⊆ (N × N)

This proof depends on syntax axioms:   ⊆ wss 3895   ∩ cin 4089   E cep 5549   × cxp 5648  Ncnpi 10932   <_N clti 10935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-in 4091  df-lti 10963

This theorem is used by:  ltapi  10991  ltmpi  10992  nlt1pi  10994  indpi  10995  ordpipq  11030  ltsonq  11057  archnq  11068