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Theorem bj-vprc 10016
 Description: vprc 3888 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 10015 . . 3 ¬ ∃𝑥𝑦 𝑦𝑥
2 vex 2560 . . . . . . 7 𝑦 ∈ V
32tbt 236 . . . . . 6 (𝑦𝑥 ↔ (𝑦𝑥𝑦 ∈ V))
43albii 1359 . . . . 5 (∀𝑦 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
5 dfcleq 2034 . . . . 5 (𝑥 = V ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
64, 5bitr4i 176 . . . 4 (∀𝑦 𝑦𝑥𝑥 = V)
76exbii 1496 . . 3 (∃𝑥𝑦 𝑦𝑥 ↔ ∃𝑥 𝑥 = V)
81, 7mtbi 595 . 2 ¬ ∃𝑥 𝑥 = V
9 isset 2561 . 2 (V ∈ V ↔ ∃𝑥 𝑥 = V)
108, 9mtbir 596 1 ¬ V ∈ V
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 98  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022  ax-bdn 9937  ax-bdel 9941  ax-bdsep 10004 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559 This theorem is referenced by:  bj-nvel  10017  bj-vnex  10018  bj-intexr  10028  bj-intnexr  10029
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