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Theorem ipo0 44446
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 2011 . . . . 5 𝑥 = 𝑥
2 vex 3483 . . . . . 6 𝑥 ∈ V
32ideq 5861 . . . . 5 (𝑥 I 𝑥𝑥 = 𝑥)
41, 3mpbir 231 . . . 4 𝑥 I 𝑥
5 poirr 5602 . . . . 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 4406 . 2 ( I Po 𝐴𝐴 = ∅)
9 po0 5607 . . 3 I Po ∅
10 poeq2 5594 . . 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 1540  wcel 2108  c0 4332   class class class wbr 5141   I cid 5575   Po wpo 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5142  df-opab 5204  df-id 5576  df-po 5590  df-xp 5689  df-rel 5690
This theorem is referenced by: (None)
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