Theorem bj-ru 36995

Description: Remove dependency on ax-13 2372 (and df-v 3438) from Russell's paradox ru 3734 expressed with primitive symbols and with a class variable 𝑉. Note the more economical use of elissetv 2812 instead of isset 3450 to avoid use of df-v 3438. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ru	⊢ ¬ {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉

Proof of Theorem bj-ru

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-ru1 36994	. 2 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}
2		elissetv 2812	. 2 ⊢ ({𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉 → ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥})
3	1, 2	mto 197	1 ⊢ ¬ {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉

Syntax hints:  ¬ wn 3   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806

This theorem is referenced by: (None)