Theorem bj-vprc 16612

Description: vprc 4226 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-vprc	⊢ ¬ V ∈ V

Proof of Theorem bj-vprc

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-nalset 16611	. . 3 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2		vex 2806	. . . . . . 7 ⊢ 𝑦 ∈ V
3	2	tbt 247	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
4	3	albii 1519	. . . . 5 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
5		dfcleq 2225	. . . . 5 ⊢ (𝑥 = V ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
6	4, 5	bitr4i 187	. . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V)
7	6	exbii 1654	. . 3 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V)
8	1, 7	mtbi 677	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
9		isset 2810	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
10	8, 9	mtbir 678	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   ↔ wb 105  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-bdn 16533  ax-bdel 16537  ax-bdsep 16600

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2805

This theorem is referenced by:  bj-nvel  16613  bj-vnex  16614  bj-intexr  16624  bj-intnexr  16625