Theorem sotr 4303

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 4298	. 2 |- ( R Or A -> R Po A )
2		potr 4293	. 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 281	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 103   /\ w3a 973   e. wcel 2141   class class class wbr 3989   Po wpo 4279   Or wor 4280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-po 4281  df-iso 4282

This theorem is referenced by:  sotri  5006  cauappcvgprlemdisj  7613  suplocexprlemru  7681