Theorem ltrelpi 7544

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 7527	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 3428	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3259	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3199   ⊆ wss 3200   E cep 4384   × cxp 4723  Ncnpi 7492   <_N clti 7495

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-lti 7527

This theorem is referenced by:  ltsonq  7618  caucvgprlemk  7885  caucvgprlem1  7899  caucvgprlem2  7900  caucvgprprlemk  7903  caucvgprprlemval  7908  caucvgprprlem1  7929  caucvgprprlem2  7930  ltrenn  8075