Theorem ltrelpi 7644

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 7627	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 3444	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3272	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3212   ⊆ wss 3213   E cep 4410   × cxp 4749  Ncnpi 7592   <_N clti 7595

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3219  df-ss 3226  df-lti 7627

This theorem is referenced by:  ltsonq  7718  caucvgprlemk  7985  caucvgprlem1  7999  caucvgprlem2  8000  caucvgprprlemk  8003  caucvgprprlemval  8008  caucvgprprlem1  8029  caucvgprprlem2  8030  ltrenn  8175