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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

Proof of Theorem ipo0
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 equid 1688 . . . . 5
2 vex 2951 . . . . . 6
32ideq 5017 . . . . 5
41, 3mpbir 201 . . . 4
5 poirr 4506 . . . . 5
65ex 424 . . . 4
74, 6mt2i 112 . . 3
87eq0rdv 3654 . 2
9 po0 4510 . . 3
10 poeq2 4499 . . 3
119, 10mpbiri 225 . 2
128, 11impbii 181 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 177   wceq 1652   wcel 1725  c0 3620   class class class wbr 4204   cid 4485   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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