Theorem ltrelpi 7685

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 7668	. 2 ⊢ <N = ( E ∩ (N × N))
2		inss2 3452	. 2 ⊢ ( E ∩ (N × N)) ⊆ (N × N)
3	1, 2	eqsstri 3280	1 ⊢ <N ⊆ (N × N)

Syntax hints:   ∩ cin 3219   ⊆ wss 3220   E cep 4430   × cxp 4770  Ncnpi 7633   <_N clti 7636

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-lti 7668

This theorem is referenced by:  ltsonq  7759  caucvgprlemk  8026  caucvgprlem1  8040  caucvgprlem2  8041  caucvgprprlemk  8044  caucvgprprlemval  8049  caucvgprprlem1  8070  caucvgprprlem2  8071  ltrenn  8216