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Theorem ipo0 27619
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 1688 . . . . 5  |-  x  =  x
2 vex 2951 . . . . . 6  |-  x  e. 
_V
32ideq 5017 . . . . 5  |-  ( x  _I  x  <->  x  =  x )
41, 3mpbir 201 . . . 4  |-  x  _I  x
5 poirr 4506 . . . . 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 3654 . 2  |-  (  _I  Po  A  ->  A  =  (/) )
9 po0 4510 . . 3  |-  _I  Po  (/)
10 poeq2 4499 . . 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 1652    e. wcel 1725   (/)c0 3620   class class class wbr 4204    _I cid 4485    Po wpo 4493
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-id 4490  df-po 4495  df-xp 4876  df-rel 4877
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