Theorem bj-ru0 34248

Description: The FOL part of Russell's paradox ru 3770 (see also bj-ru1 34249, bj-ru 34250). Use of elequ1 2117, bj-elequ12 34007 (instead of eleq1 2900, eleq12d 2907 as in ru 3770) permits to remove dependency on ax-10 2141, ax-11 2157, ax-12 2173, ax-ext 2793, df-sb 2066, df-clab 2800, df-cleq 2814, df-clel 2893. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-ru0

Step	Hyp	Ref	Expression
1		pm5.19 390	. 2 ⊢ ¬ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)
2		elequ1 2117	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ↔ 𝑦 ∈ 𝑦))
3		bj-elequ12 34007	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
4	3	anidms 569	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
5	4	notbid 320	. . . 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 1999	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥) → (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦))
8	1, 7	mto 199	1 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥)

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1531

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777

This theorem is referenced by:  bj-ru1  34249