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Theorem ipo0 44411
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 𝐴𝐴 = ∅)

Proof of Theorem ipo0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 equid 2007 . . . . 5 𝑥 = 𝑥
2 vex 3481 . . . . . 6 𝑥 ∈ V
32ideq 5861 . . . . 5 (𝑥 I 𝑥𝑥 = 𝑥)
41, 3mpbir 231 . . . 4 𝑥 I 𝑥
5 poirr 5604 . . . . 5 (( I Po 𝐴𝑥𝐴) → ¬ 𝑥 I 𝑥)
65ex 412 . . . 4 ( I Po 𝐴 → (𝑥𝐴 → ¬ 𝑥 I 𝑥))
74, 6mt2i 137 . . 3 ( I Po 𝐴 → ¬ 𝑥𝐴)
87eq0rdv 4413 . 2 ( I Po 𝐴𝐴 = ∅)
9 po0 5609 . . 3 I Po ∅
10 poeq2 5595 . . 3 (𝐴 = ∅ → ( I Po 𝐴 ↔ I Po ∅))
119, 10mpbiri 258 . 2 (𝐴 = ∅ → I Po 𝐴)
128, 11impbii 209 1 ( I Po 𝐴𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1535  wcel 2104  c0 4339   class class class wbr 5150   I cid 5576   Po wpo 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5151  df-opab 5213  df-id 5577  df-po 5591  df-xp 5690  df-rel 5691
This theorem is referenced by: (None)
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