Theorem exnel 36016

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 9514	. 2 ⊢ ¬ 𝑦 ∈ 𝑦
2	1	nfth 1803	. . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝑦
3		ax8 2120	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑦))
4	3	con3d 152	. . 3 ⊢ (𝑥 = 𝑦 → (¬ 𝑦 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑦))
5	2, 4	spime 2394	. 2 ⊢ (¬ 𝑦 ∈ 𝑦 → ∃𝑥 ¬ 𝑥 ∈ 𝑦)
6	1, 5	ax-mp 5	1 ⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦

Syntax hints:  ¬ wn 3  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-13 2377  ax-sep 5243  ax-pr 5379  ax-reg 9509

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786

This theorem is referenced by: (None)