Theorem exnel 35626

Description: There is always a set not in 𝑦. (Contributed by Scott Fenton, 13-Dec-2010.)

Assertion

Ref	Expression
exnel	⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦

Proof of Theorem exnel

Step	Hyp	Ref	Expression
1		elirrv 9639	. 2 ⊢ ¬ 𝑦 ∈ 𝑦
2	1	nfth 1796	. . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝑦
3		ax8 2105	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑦))
4	3	con3d 152	. . 3 ⊢ (𝑥 = 𝑦 → (¬ 𝑦 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑦))
5	2, 4	spime 2383	. 2 ⊢ (¬ 𝑦 ∈ 𝑦 → ∃𝑥 ¬ 𝑥 ∈ 𝑦)
6	1, 5	ax-mp 5	1 ⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦

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 2366  ax-ext 2697  ax-sep 5304  ax-pr 5433  ax-reg 9635

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-v 3464  df-un 3952  df-sn 4634  df-pr 4636

This theorem is referenced by: (None)