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Theorem ipo0 42821
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 2016 . . . . 5 𝑥 = 𝑥
2 vex 3451 . . . . . 6 𝑥 ∈ V
32ideq 5812 . . . . 5 (𝑥 I 𝑥𝑥 = 𝑥)
41, 3mpbir 230 . . . 4 𝑥 I 𝑥
5 poirr 5561 . . . . 5 (( I Po 𝐴𝑥𝐴) → ¬ 𝑥 I 𝑥)
65ex 414 . . . 4 ( I Po 𝐴 → (𝑥𝐴 → ¬ 𝑥 I 𝑥))
74, 6mt2i 137 . . 3 ( I Po 𝐴 → ¬ 𝑥𝐴)
87eq0rdv 4368 . 2 ( I Po 𝐴𝐴 = ∅)
9 po0 5566 . . 3 I Po ∅
10 poeq2 5553 . . 3 (𝐴 = ∅ → ( I Po 𝐴 ↔ I Po ∅))
119, 10mpbiri 258 . 2 (𝐴 = ∅ → I Po 𝐴)
128, 11impbii 208 1 ( I Po 𝐴𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1542  wcel 2107  c0 4286   class class class wbr 5109   I cid 5534   Po wpo 5547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-id 5535  df-po 5549  df-xp 5643  df-rel 5644
This theorem is referenced by: (None)
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