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Theorem bj-vprc 14618
Description: vprc 4135 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.
StepHypRef Expression
1 bj-nalset 14617 . . 3 ¬ ∃𝑥𝑦 𝑦𝑥
2 vex 2740 . . . . . . 7 𝑦 ∈ V
32tbt 247 . . . . . 6 (𝑦𝑥 ↔ (𝑦𝑥𝑦 ∈ V))
43albii 1470 . . . . 5 (∀𝑦 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
5 dfcleq 2171 . . . . 5 (𝑥 = V ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
64, 5bitr4i 187 . . . 4 (∀𝑦 𝑦𝑥𝑥 = V)
76exbii 1605 . . 3 (∃𝑥𝑦 𝑦𝑥 ↔ ∃𝑥 𝑥 = V)
81, 7mtbi 670 . 2 ¬ ∃𝑥 𝑥 = V
9 isset 2743 . 2 (V ∈ V ↔ ∃𝑥 𝑥 = V)
108, 9mtbir 671 1 ¬ V ∈ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 105  wal 1351   = wceq 1353  wex 1492  wcel 2148  Vcvv 2737
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 14539  ax-bdel 14543  ax-bdsep 14606
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 2739
This theorem is referenced by:  bj-nvel  14619  bj-vnex  14620  bj-intexr  14630  bj-intnexr  14631
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