Theorem sotr 4421

Description: A strict order relation is a transitive relation. (Contributed by NM, 21-Jan-1996.)

Assertion

Ref	Expression
sotr	|- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( ( B R C /\ C R D ) -> B R D ) )

Proof of Theorem sotr

Step	Hyp	Ref	Expression
1		sopo 4416	. 2 |- ( R Or A -> R Po A )
2		potr 4411	. 2 |- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( ( B R C /\ C R D ) -> B R D ) )
3	1, 2	sylan 283	1 |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( ( B R C /\ C R D ) -> B R D ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   e. wcel 2202   class class class wbr 4093   Po wpo 4397   Or wor 4398

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-in1 619  ax-in2 620  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 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-po 4399  df-iso 4400

This theorem is referenced by:  sotri  5139  cauappcvgprlemdisj  7931  suplocexprlemru  7999