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Theorem istsr2 14578
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 14577 . 2  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( X  X.  X
)  C_  ( R  u.  `' R ) ) )
3 qfto 5196 . . 3  |-  ( ( X  X.  X ) 
C_  ( R  u.  `' R )  <->  A. x  e.  X  A. y  e.  X  ( x R y  \/  y R x ) )
43anbi2i 676 . 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 241 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 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650    u. cun 3262    C_ wss 3264   class class class wbr 4154    X. cxp 4817   `'ccnv 4818   dom cdm 4819   PosetRelcps 14552    TosetRel ctsr 14553
This theorem is referenced by:  tsrlin  14579  tsrss  14583
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 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-cnv 4827  df-dm 4829  df-tsr 14558
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