Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ipo0 Structured version   Visualization version   GIF version

Theorem ipo0 41073
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 2020 . . . . 5 𝑥 = 𝑥
2 vex 3483 . . . . . 6 𝑥 ∈ V
32ideq 5710 . . . . 5 (𝑥 I 𝑥𝑥 = 𝑥)
41, 3mpbir 234 . . . 4 𝑥 I 𝑥
5 poirr 5472 . . . . 5 (( I Po 𝐴𝑥𝐴) → ¬ 𝑥 I 𝑥)
65ex 416 . . . 4 ( I Po 𝐴 → (𝑥𝐴 → ¬ 𝑥 I 𝑥))
74, 6mt2i 139 . . 3 ( I Po 𝐴 → ¬ 𝑥𝐴)
87eq0rdv 4340 . 2 ( I Po 𝐴𝐴 = ∅)
9 po0 5477 . . 3 I Po ∅
10 poeq2 5465 . . 3 (𝐴 = ∅ → ( I Po 𝐴 ↔ I Po ∅))
119, 10mpbiri 261 . 2 (𝐴 = ∅ → I Po 𝐴)
128, 11impbii 212 1 ( I Po 𝐴𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209   = wceq 1538  wcel 2115  c0 4276   class class class wbr 5052   I cid 5446   Po wpo 5459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-id 5447  df-po 5461  df-xp 5548  df-rel 5549
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator