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Theorem exnel 33055
 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 9038 . 2 ¬ 𝑦𝑦
21nfth 1802 . . 3 𝑥 ¬ 𝑦𝑦
3 ax8 2120 . . . 4 (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
43con3d 155 . . 3 (𝑥 = 𝑦 → (¬ 𝑦𝑦 → ¬ 𝑥𝑦))
52, 4spime 2407 . 2 𝑦𝑦 → ∃𝑥 ¬ 𝑥𝑦)
61, 5ax-mp 5 1 𝑥 ¬ 𝑥𝑦
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3  ∃wex 1780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306  ax-reg 9034 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-v 3475  df-dif 3916  df-un 3918  df-nul 4270  df-sn 4544  df-pr 4546 This theorem is referenced by: (None)
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