Theorem bj-ru 37779

Description: Remove dependency on ax-13 2401 (and df-v 3452) from Russell's paradox ru 3737 expressed with primitive symbols and with a class variable 𝑉. Note the more economical use of elissetv 2841 instead of isset 3464 to avoid use of df-v 3452. (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 37778	. 2 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}
2		elissetv 2841	. 2 ⊢ ({𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉 → ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥})
3	1, 2	mto 200	1 ⊢ ¬ {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝑉

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835

This theorem is used by: (None)