Theorem ltrelpi 7519

Description: Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)

Assertion

Ref	Expression
ltrelpi	⊢ <N ⊆ (N × N)

Proof of Theorem ltrelpi

Step	Hyp	Ref	Expression
1		df-lti 7502	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 3425	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3256	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3196   ⊆ wss 3197   E cep 4378   × cxp 4717  Ncnpi 7467   <_N clti 7470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-lti 7502

This theorem is referenced by:  ltsonq  7593  caucvgprlemk  7860  caucvgprlem1  7874  caucvgprlem2  7875  caucvgprprlemk  7878  caucvgprprlemval  7883  caucvgprprlem1  7904  caucvgprprlem2  7905  ltrenn  8050