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Theorem bj-vprc 13417
 Description: vprc 4092 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 13416 . . 3 ¬ ∃𝑥𝑦 𝑦𝑥
2 vex 2712 . . . . . . 7 𝑦 ∈ V
32tbt 246 . . . . . 6 (𝑦𝑥 ↔ (𝑦𝑥𝑦 ∈ V))
43albii 1447 . . . . 5 (∀𝑦 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
5 dfcleq 2148 . . . . 5 (𝑥 = V ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
64, 5bitr4i 186 . . . 4 (∀𝑦 𝑦𝑥𝑥 = V)
76exbii 1582 . . 3 (∃𝑥𝑦 𝑦𝑥 ↔ ∃𝑥 𝑥 = V)
81, 7mtbi 660 . 2 ¬ ∃𝑥 𝑥 = V
9 isset 2715 . 2 (V ∈ V ↔ ∃𝑥 𝑥 = V)
108, 9mtbir 661 1 ¬ V ∈ V
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 104  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 2125  Vcvv 2709 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-13 2127  ax-14 2128  ax-ext 2136  ax-bdn 13338  ax-bdel 13342  ax-bdsep 13405 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-v 2711 This theorem is referenced by:  bj-nvel  13418  bj-vnex  13419  bj-intexr  13429  bj-intnexr  13430
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