Theorem ltrelpi 10310

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 10296	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 4205	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 4000	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3934   ⊆ wss 3935   E cep 5463   × cxp 5552  Ncnpi 10265   <_N clti 10268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-in 3942  df-ss 3951  df-lti 10296

This theorem is referenced by:  ltapi  10324  ltmpi  10325  nlt1pi  10327  indpi  10328  ordpipq  10363  ltsonq  10390  archnq  10401