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Theorem bj-vprc 10389
Description: vprc 3915 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 10388 . . 3 ¬ ∃𝑥𝑦 𝑦𝑥
2 vex 2577 . . . . . . 7 𝑦 ∈ V
32tbt 240 . . . . . 6 (𝑦𝑥 ↔ (𝑦𝑥𝑦 ∈ V))
43albii 1375 . . . . 5 (∀𝑦 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
5 dfcleq 2050 . . . . 5 (𝑥 = V ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
64, 5bitr4i 180 . . . 4 (∀𝑦 𝑦𝑥𝑥 = V)
76exbii 1512 . . 3 (∃𝑥𝑦 𝑦𝑥 ↔ ∃𝑥 𝑥 = V)
81, 7mtbi 605 . 2 ¬ ∃𝑥 𝑥 = V
9 isset 2578 . 2 (V ∈ V ↔ ∃𝑥 𝑥 = V)
108, 9mtbir 606 1 ¬ V ∈ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 102  wal 1257   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038  ax-bdn 10310  ax-bdel 10314  ax-bdsep 10377
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  bj-nvel  10390  bj-vnex  10391  bj-intexr  10401  bj-intnexr  10402
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