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

Theorem istsr2 14343
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 14342 . 2  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( X  X.  X
)  C_  ( R  u.  `' R ) ) )
3 qfto 5080 . . 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 1632    e. wcel 1696   A.wral 2556    u. cun 3163    C_ wss 3165   class class class wbr 4039    X. cxp 4703   `'ccnv 4704   dom cdm 4705   PosetRelcps 14317    TosetRel ctsr 14318
This theorem is referenced by:  tsrlin  14344  tsrss  14348  tolat  25389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-tsr 14323
  Copyright terms: Public domain W3C validator