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Theorem ipo0 39147
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 2086 . . . . 5 𝑥 = 𝑥
2 vex 3335 . . . . . 6 𝑥 ∈ V
32ideq 5422 . . . . 5 (𝑥 I 𝑥𝑥 = 𝑥)
41, 3mpbir 221 . . . 4 𝑥 I 𝑥
5 poirr 5190 . . . . 5 (( I Po 𝐴𝑥𝐴) → ¬ 𝑥 I 𝑥)
65ex 449 . . . 4 ( I Po 𝐴 → (𝑥𝐴 → ¬ 𝑥 I 𝑥))
74, 6mt2i 132 . . 3 ( I Po 𝐴 → ¬ 𝑥𝐴)
87eq0rdv 4114 . 2 ( I Po 𝐴𝐴 = ∅)
9 po0 5194 . . 3 I Po ∅
10 poeq2 5183 . . 3 (𝐴 = ∅ → ( I Po 𝐴 ↔ I Po ∅))
119, 10mpbiri 248 . 2 (𝐴 = ∅ → I Po 𝐴)
128, 11impbii 199 1 ( I Po 𝐴𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1624  wcel 2131  c0 4050   class class class wbr 4796   I cid 5165   Po wpo 5177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-id 5166  df-po 5179  df-xp 5264  df-rel 5265
This theorem is referenced by: (None)
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