Theorem bj-vprc 14651

Description: vprc 4136 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 14650	. . 3 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2		vex 2741	. . . . . . 7 ⊢ 𝑦 ∈ V
3	2	tbt 247	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
4	3	albii 1470	. . . . 5 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
5		dfcleq 2171	. . . . 5 ⊢ (𝑥 = V ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
6	4, 5	bitr4i 187	. . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V)
7	6	exbii 1605	. . 3 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V)
8	1, 7	mtbi 670	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
9		isset 2744	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
10	8, 9	mtbir 671	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   ↔ wb 105  ∀wal 1351   = wceq 1353  ∃wex 1492   ∈ wcel 2148  Vcvv 2738

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 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-13 2150  ax-14 2151  ax-ext 2159  ax-bdn 14572  ax-bdel 14576  ax-bdsep 14639

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2740

This theorem is referenced by:  bj-nvel  14652  bj-vnex  14653  bj-intexr  14663  bj-intnexr  14664