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Theorem psasym 14562
Description: A poset is antisymmetric. (Contributed by NM, 12-May-2008.)
Assertion
Ref Expression
psasym  |-  ( ( R  e.  PosetRel  /\  A R B  /\  B R A )  ->  A  =  B )

Proof of Theorem psasym
StepHypRef Expression
1 pslem 14558 . . 3  |-  ( R  e.  PosetRel  ->  ( ( ( A R B  /\  B R A )  ->  A R A )  /\  ( A  e.  U. U. R  ->  A R A )  /\  ( ( A R B  /\  B R A )  ->  A  =  B )
) )
21simp3d 971 . 2  |-  ( R  e.  PosetRel  ->  ( ( A R B  /\  B R A )  ->  A  =  B ) )
323impib 1151 1  |-  ( ( R  e.  PosetRel  /\  A R B  /\  B R A )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   U.cuni 3950   class class class wbr 4146   PosetRelcps 14544
This theorem is referenced by:  psss  14566  spwmo  14578  ordtt1  17358  ordthauslem  17362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-res 4823  df-ps 14549
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