Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  exnel Structured version   Visualization version   GIF version

Theorem exnel 35296
Description: There is always a set not in 𝑦. (Contributed by Scott Fenton, 13-Dec-2010.)
Assertion
Ref Expression
exnel 𝑥 ¬ 𝑥𝑦

Proof of Theorem exnel
StepHypRef Expression
1 elirrv 9588 . 2 ¬ 𝑦𝑦
21nfth 1795 . . 3 𝑥 ¬ 𝑦𝑦
3 ax8 2104 . . . 4 (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
43con3d 152 . . 3 (𝑥 = 𝑦 → (¬ 𝑦𝑦 → ¬ 𝑥𝑦))
52, 4spime 2380 . 2 𝑦𝑦 → ∃𝑥 ¬ 𝑥𝑦)
61, 5ax-mp 5 1 𝑥 ¬ 𝑥𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-13 2363  ax-ext 2695  ax-sep 5290  ax-pr 5418  ax-reg 9584
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-v 3468  df-un 3946  df-sn 4622  df-pr 4624
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator