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Theorem ipo0 27313
Description: If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ipo0  |-  (  _I  Po  A  <->  A  =  (/) )

Proof of Theorem ipo0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 equid 1683 . . . . 5  |-  x  =  x
2 vex 2895 . . . . . 6  |-  x  e. 
_V
32ideq 4958 . . . . 5  |-  ( x  _I  x  <->  x  =  x )
41, 3mpbir 201 . . . 4  |-  x  _I  x
5 poirr 4448 . . . . 5  |-  ( (  _I  Po  A  /\  x  e.  A )  ->  -.  x  _I  x
)
65ex 424 . . . 4  |-  (  _I  Po  A  ->  (
x  e.  A  ->  -.  x  _I  x
) )
74, 6mt2i 112 . . 3  |-  (  _I  Po  A  ->  -.  x  e.  A )
87eq0rdv 3598 . 2  |-  (  _I  Po  A  ->  A  =  (/) )
9 po0 4452 . . 3  |-  _I  Po  (/)
10 poeq2 4441 . . 3  |-  ( A  =  (/)  ->  (  _I  Po  A  <->  _I  Po  (/) ) )
119, 10mpbiri 225 . 2  |-  ( A  =  (/)  ->  _I  Po  A )
128, 11impbii 181 1  |-  (  _I  Po  A  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1649    e. wcel 1717   (/)c0 3564   class class class wbr 4146    _I cid 4427    Po wpo 4435
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-br 4147  df-opab 4201  df-id 4432  df-po 4437  df-xp 4817  df-rel 4818
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