Theorem ru0 2164

Description: The FOL statement used in the standard proof of Russell's paradox ru 3746. (Contributed by NM, 7-Aug-1994.) Extract from proof of ru 3746 and reduce axiom usage. (Revised by BJ, 12-Oct-2019.)

Assertion

Ref	Expression
ru0	⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥)

Distinct variable group:   𝑥,𝑦

Proof of Theorem ru0

Step	Hyp	Ref	Expression
1		pm5.19 390	. 2 ⊢ ¬ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)
2		elequ1 2152	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ↔ 𝑦 ∈ 𝑦))
3		elequ12 2163	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
4	3	anidms 576	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
5	4	notbid 321	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
6	2, 5	bibi12d 348	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥) ↔ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)))
7	6	spvv 2011	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥) → (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦))
8	1, 7	mto 200	1 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥)

Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by:  ru  3746  bj-ru1  37440