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Theorem exnel 34416
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 9539 . 2 ¬ 𝑦𝑦
21nfth 1804 . . 3 𝑥 ¬ 𝑦𝑦
3 ax8 2113 . . . 4 (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
43con3d 152 . . 3 (𝑥 = 𝑦 → (¬ 𝑦𝑦 → ¬ 𝑥𝑦))
52, 4spime 2388 . 2 𝑦𝑦 → ∃𝑥 ¬ 𝑥𝑦)
61, 5ax-mp 5 1 𝑥 ¬ 𝑥𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-13 2371  ax-ext 2708  ax-sep 5261  ax-pr 5389  ax-reg 9535
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-v 3450  df-un 3920  df-sn 4592  df-pr 4594
This theorem is referenced by: (None)
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