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Theorem ipo0 40658
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 2010 . . . . 5 𝑥 = 𝑥
2 vex 3495 . . . . . 6 𝑥 ∈ V
32ideq 5716 . . . . 5 (𝑥 I 𝑥𝑥 = 𝑥)
41, 3mpbir 232 . . . 4 𝑥 I 𝑥
5 poirr 5478 . . . . 5 (( I Po 𝐴𝑥𝐴) → ¬ 𝑥 I 𝑥)
65ex 413 . . . 4 ( I Po 𝐴 → (𝑥𝐴 → ¬ 𝑥 I 𝑥))
74, 6mt2i 139 . . 3 ( I Po 𝐴 → ¬ 𝑥𝐴)
87eq0rdv 4354 . 2 ( I Po 𝐴𝐴 = ∅)
9 po0 5483 . . 3 I Po ∅
10 poeq2 5471 . . 3 (𝐴 = ∅ → ( I Po 𝐴 ↔ I Po ∅))
119, 10mpbiri 259 . 2 (𝐴 = ∅ → I Po 𝐴)
128, 11impbii 210 1 ( I Po 𝐴𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207   = wceq 1528  wcel 2105  c0 4288   class class class wbr 5057   I cid 5452   Po wpo 5465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-po 5467  df-xp 5554  df-rel 5555
This theorem is referenced by: (None)
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