Theorem pstr 17211

Description: A poset is transitive. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
pstr	|- ( ( R e. PosetRel /\ A R B /\ B R C ) -> A R C )

Proof of Theorem pstr

Step	Hyp	Ref	Expression
1		pslem 17206	. . 3 |- ( R e. PosetRel -> ( ( ( A R B /\ B R C ) -> A R C ) /\ ( A e. U. U. R -> A R A ) /\ ( ( A R B /\ B R A ) -> A = B ) ) )
2	1	simp1d 1073	. 2 |- ( R e. PosetRel -> ( ( A R B /\ B R C ) -> A R C ) )
3	2	3impib 1262	1 |- ( ( R e. PosetRel /\ A R B /\ B R C ) -> A R C )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   U.cuni 4436   class class class wbr 4653   PosetRelcps 17198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-res 5126  df-ps 17200

This theorem is referenced by:  tsrlemax  17220