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Theorem exnel 35398
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 9619 . 2 ¬ 𝑦𝑦
21nfth 1796 . . 3 𝑥 ¬ 𝑦𝑦
3 ax8 2105 . . . 4 (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
43con3d 152 . . 3 (𝑥 = 𝑦 → (¬ 𝑦𝑦 → ¬ 𝑥𝑦))
52, 4spime 2384 . 2 𝑦𝑦 → ∃𝑥 ¬ 𝑥𝑦)
61, 5ax-mp 5 1 𝑥 ¬ 𝑥𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wex 1774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-13 2367  ax-ext 2699  ax-sep 5299  ax-pr 5429  ax-reg 9615
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-v 3473  df-un 3952  df-sn 4630  df-pr 4632
This theorem is referenced by: (None)
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