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Theorem ipo0 36439
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 1842 . . . . 5  |-  x  =  x
2 vex 3090 . . . . . 6  |-  x  e. 
_V
32ideq 5007 . . . . 5  |-  ( x  _I  x  <->  x  =  x )
41, 3mpbir 212 . . . 4  |-  x  _I  x
5 poirr 4786 . . . . 5  |-  ( (  _I  Po  A  /\  x  e.  A )  ->  -.  x  _I  x
)
65ex 435 . . . 4  |-  (  _I  Po  A  ->  (
x  e.  A  ->  -.  x  _I  x
) )
74, 6mt2i 121 . . 3  |-  (  _I  Po  A  ->  -.  x  e.  A )
87eq0rdv 3803 . 2  |-  (  _I  Po  A  ->  A  =  (/) )
9 po0 4790 . . 3  |-  _I  Po  (/)
10 poeq2 4779 . . 3  |-  ( A  =  (/)  ->  (  _I  Po  A  <->  _I  Po  (/) ) )
119, 10mpbiri 236 . 2  |-  ( A  =  (/)  ->  _I  Po  A )
128, 11impbii 190 1  |-  (  _I  Po  A  <->  A  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    = wceq 1437    e. wcel 1870   (/)c0 3767   class class class wbr 4426    _I cid 4764    Po wpo 4773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-id 4769  df-po 4775  df-xp 4860  df-rel 4861
This theorem is referenced by: (None)
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