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Theorem istsr2 14638
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 14637 . 2  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( X  X.  X
)  C_  ( R  u.  `' R ) ) )
3 qfto 5246 . . 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 1652    e. wcel 1725   A.wral 2697    u. cun 3310    C_ wss 3312   class class class wbr 4204    X. cxp 4867   `'ccnv 4868   dom cdm 4869   PosetRelcps 14612    TosetRel ctsr 14613
This theorem is referenced by:  tsrlin  14639  tsrss  14643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-cnv 4877  df-dm 4879  df-tsr 14618
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