Theorem sotr 4811

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 4806	. 2 |- ( R Or A -> R Po A )
2		potr 4801	. 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 469	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 367   /\ w3a 971   e. wcel 1823   class class class wbr 4439   Po wpo 4787   Or wor 4788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-po 4789  df-so 4790

This theorem is referenced by:  sotr2  4818  wetrep  4861  wereu2  4865  sotri  5382  suplub2  7912  slttr  29674  fin2solem  30282