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Theorem ipo0 31228
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 1740 . . . . 5  |-  x  =  x
2 vex 3121 . . . . . 6  |-  x  e. 
_V
32ideq 5160 . . . . 5  |-  ( x  _I  x  <->  x  =  x )
41, 3mpbir 209 . . . 4  |-  x  _I  x
5 poirr 4816 . . . . 5  |-  ( (  _I  Po  A  /\  x  e.  A )  ->  -.  x  _I  x
)
65ex 434 . . . 4  |-  (  _I  Po  A  ->  (
x  e.  A  ->  -.  x  _I  x
) )
74, 6mt2i 118 . . 3  |-  (  _I  Po  A  ->  -.  x  e.  A )
87eq0rdv 3825 . 2  |-  (  _I  Po  A  ->  A  =  (/) )
9 po0 4820 . . 3  |-  _I  Po  (/)
10 poeq2 4809 . . 3  |-  ( A  =  (/)  ->  (  _I  Po  A  <->  _I  Po  (/) ) )
119, 10mpbiri 233 . 2  |-  ( A  =  (/)  ->  _I  Po  A )
128, 11impbii 188 1  |-  (  _I  Po  A  <->  A  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1379    e. wcel 1767   (/)c0 3790   class class class wbr 4452    _I cid 4795    Po wpo 4803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4453  df-opab 4511  df-id 4800  df-po 4805  df-xp 5010  df-rel 5011
This theorem is referenced by: (None)
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