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 32013
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 8666 . 2 ¬ 𝑦𝑦
21nfth 1876 . . 3 𝑥 ¬ 𝑦𝑦
3 ax8 2145 . . . 4 (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
43con3d 148 . . 3 (𝑥 = 𝑦 → (¬ 𝑦𝑦 → ¬ 𝑥𝑦))
52, 4spime 2401 . 2 𝑦𝑦 → ∃𝑥 ¬ 𝑥𝑦)
61, 5ax-mp 5 1 𝑥 ¬ 𝑥𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wex 1853
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-8 2141  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  ax-reg 8662
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-nul 4059  df-sn 4322  df-pr 4324
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator