MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  istsr2 Unicode version

Theorem istsr2 14327
Description: The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
Hypothesis
Ref Expression
istsr.1  |-  X  =  dom  R
Assertion
Ref Expression
istsr2  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  A. x  e.  X  A. y  e.  X  (
x R y  \/  y R x ) ) )
Distinct variable groups:    x, y, R    x, X, y

Proof of Theorem istsr2
StepHypRef Expression
1 istsr.1 . . 3  |-  X  =  dom  R
21istsr 14326 . 2  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( X  X.  X
)  C_  ( R  u.  `' R ) ) )
3 qfto 5064 . . 3  |-  ( ( X  X.  X ) 
C_  ( R  u.  `' R )  <->  A. x  e.  X  A. y  e.  X  ( x R y  \/  y R x ) )
43anbi2i 675 . 2  |-  ( ( R  e.  PosetRel  /\  ( X  X.  X )  C_  ( R  u.  `' R ) )  <->  ( R  e. 
PosetRel  /\  A. x  e.  X  A. y  e.  X  ( x R y  \/  y R x ) ) )
52, 4bitri 240 1  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  A. x  e.  X  A. y  e.  X  (
x R y  \/  y R x ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    u. cun 3150    C_ wss 3152   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   dom cdm 4689   PosetRelcps 14301    TosetRel ctsr 14302
This theorem is referenced by:  tsrlin  14328  tsrss  14332  tolat  25286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-tsr 14307
  Copyright terms: Public domain W3C validator