Theorem exnel 30952

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 8387	. 2 ⊢ ¬ 𝑦 ∈ 𝑦
2	1	nfth 1718	. . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝑦
3		ax8 1983	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑦))
4	3	con3d 147	. . 3 ⊢ (𝑥 = 𝑦 → (¬ 𝑦 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑦))
5	2, 4	spime 2244	. 2 ⊢ (¬ 𝑦 ∈ 𝑦 → ∃𝑥 ¬ 𝑥 ∈ 𝑦)
6	1, 5	ax-mp 5	1 ⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦

Syntax hints:  ¬ wn 3  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-reg 8380

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128

This theorem is referenced by: (None)