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Theorem ipo0 39155
 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 2094 . . . . 5 𝑥 = 𝑥
2 vex 3343 . . . . . 6 𝑥 ∈ V
32ideq 5430 . . . . 5 (𝑥 I 𝑥𝑥 = 𝑥)
41, 3mpbir 221 . . . 4 𝑥 I 𝑥
5 poirr 5198 . . . . 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 4122 . 2 ( I Po 𝐴𝐴 = ∅)
9 po0 5202 . . 3 I Po ∅
10 poeq2 5191 . . 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 1632   ∈ wcel 2139  ∅c0 4058   class class class wbr 4804   I cid 5173   Po wpo 5185 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-po 5187  df-xp 5272  df-rel 5273 This theorem is referenced by: (None)
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