Theorem exnel 35784

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 9634	. 2 ⊢ ¬ 𝑦 ∈ 𝑦
2	1	nfth 1798	. . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝑦
3		ax8 2112	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑦))
4	3	con3d 152	. . 3 ⊢ (𝑥 = 𝑦 → (¬ 𝑦 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑦))
5	2, 4	spime 2392	. 2 ⊢ (¬ 𝑦 ∈ 𝑦 → ∃𝑥 ¬ 𝑥 ∈ 𝑦)
6	1, 5	ax-mp 5	1 ⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦

Syntax hints:  ¬ wn 3  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-13 2375  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-reg 9630

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634

This theorem is referenced by: (None)