Theorem ltrelpi 9711

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 9697	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 3834	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3635	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3573   ⊆ wss 3574   E cep 5028   × cxp 5112  Ncnpi 9666   <_N clti 9669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-lti 9697

This theorem is referenced by:  ltapi  9725  ltmpi  9726  nlt1pi  9728  indpi  9729  ordpipq  9764  ltsonq  9791  archnq  9802