Theorem exnel 35766

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 9665	. 2 ⊢ ¬ 𝑦 ∈ 𝑦
2	1	nfth 1799	. . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝑦
3		ax8 2114	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑦))
4	3	con3d 152	. . 3 ⊢ (𝑥 = 𝑦 → (¬ 𝑦 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑦))
5	2, 4	spime 2397	. 2 ⊢ (¬ 𝑦 ∈ 𝑦 → ∃𝑥 ¬ 𝑥 ∈ 𝑦)
6	1, 5	ax-mp 5	1 ⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦

Syntax hints:  ¬ wn 3  ∃wex 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-13 2380  ax-ext 2711  ax-sep 5317  ax-pr 5447  ax-reg 9661

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-v 3490  df-un 3981  df-sn 4649  df-pr 4651

This theorem is referenced by: (None)