Theorem bj-ru 36423

Description: Remove dependency on ax-13 2367 (and df-v 3473) from Russell's paradox ru 3775 expressed with primitive symbols and with a class variable 𝑉. Note the more economical use of elissetv 2810 instead of isset 3484 to avoid use of df-v 3473. (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 36422	. 2 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}
2		elissetv 2810	. 2 ⊢ ({𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉 → ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥})
3	1, 2	mto 196	1 ⊢ ¬ {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉

Syntax hints:  ¬ wn 3   = wceq 1534  ∃wex 1774   ∈ wcel 2099  {cab 2705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806

This theorem is referenced by: (None)